Functional a posteriori error estimates for space-time isogeometric - - PowerPoint PPT Presentation

1/34 Johann Radon Institute for Computational and Applied Mathematics

Functional a posteriori error estimates for space-time isogeometric approximations

• f parabolic I-BVPs

Svetlana Matculevich1

joint work with U. Langer2 and S. Repin3

1, 2 Johann Radon Institute for Computational and Applied Mathematics, Austrian

Academy of Sciences, Austria

3 St. Petersburg V.A. Steklov Institute of Mathematics, Russia; University of Jyväskylä,

Finland

Workshop 2: November 07-11, 2016 Space-Time Methods for PDEs

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

2/34 Johann Radon Institute for Computational and Applied Mathematics

Functional type error estimates

For a class of parabolic I-BVP problems ∂tu + L u = f , u(0) = u0, in Ω ⊂ Rd, t ∈ (0, T) u = 0

• n ∂Ω,

with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M(v, D(Ω, u0, f )) ≤ | | |u − v| | | ≤ M(v, D(Ω, u0, f )), minorant error majorant V u v M M universal for any v ∈ V , computable, reliable, i.e., | | |u − v| | | ≤ M(v, D), realistic in comparison to error, i.e., Ieff =

M | | |u−v| | | is close to 1,

consistent, i.e., M(v) is continuous with respect to v and M(u) = 0, efficient for adaptive algorithms Vh → Vhref .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

2/34 Johann Radon Institute for Computational and Applied Mathematics

Functional type error estimates

For a class of parabolic I-BVP problems ∂tu + L u = f , u(0) = u0, in Ω ⊂ Rd, t ∈ (0, T) u = 0

• n ∂Ω,

with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M(v, D(Ω, u0, f )) ≤ | | |u − v| | | ≤ M(v, D(Ω, u0, f )), minorant error majorant V u v M M universal for any v ∈ V , computable, reliable, i.e., | | |u − v| | | ≤ M(v, D), realistic in comparison to error, i.e., Ieff =

M | | |u−v| | | is close to 1,

consistent, i.e., M(v) is continuous with respect to v and M(u) = 0, efficient for adaptive algorithms Vh → Vhref .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

2/34 Johann Radon Institute for Computational and Applied Mathematics

Functional type error estimates

For a class of parabolic I-BVP problems ∂tu + L u = f , u(0) = u0, in Ω ⊂ Rd, t ∈ (0, T) u = 0

• n ∂Ω,

with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M(v, D(Ω, u0, f )) ≤ | | |u − v| | | ≤ M(v, D(Ω, u0, f )), minorant error majorant V u v M M universal for any v ∈ V , computable, reliable, i.e., | | |u − v| | | ≤ M(v, D), realistic in comparison to error, i.e., Ieff =

M | | |u−v| | | is close to 1,

consistent, i.e., M(v) is continuous with respect to v and M(u) = 0, efficient for adaptive algorithms Vh → Vhref .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

2/34 Johann Radon Institute for Computational and Applied Mathematics

Functional type error estimates

For a class of parabolic I-BVP problems ∂tu + L u = f , u(0) = u0, in Ω ⊂ Rd, t ∈ (0, T) u = 0

• n ∂Ω,

with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M(v, D(Ω, u0, f )) ≤ | | |u − v| | | ≤ M(v, D(Ω, u0, f )), minorant error majorant V u v M M universal for any v ∈ V , computable, reliable, i.e., | | |u − v| | | ≤ M(v, D), realistic in comparison to error, i.e., Ieff =

M | | |u−v| | | is close to 1,

consistent, i.e., M(v) is continuous with respect to v and M(u) = 0, efficient for adaptive algorithms Vh → Vhref .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

2/34 Johann Radon Institute for Computational and Applied Mathematics

Functional type error estimates

For a class of parabolic I-BVP problems ∂tu + L u = f , u(0) = u0, in Ω ⊂ Rd, t ∈ (0, T) u = 0

• n ∂Ω,

with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M(v, D(Ω, u0, f )) ≤ | | |u − v| | | ≤ M(v, D(Ω, u0, f )), minorant error majorant V u v M M universal for any v ∈ V , computable, reliable, i.e., | | |u − v| | | ≤ M(v, D), realistic in comparison to error, i.e., Ieff =

M | | |u−v| | | is close to 1,

consistent, i.e., M(v) is continuous with respect to v and M(u) = 0, efficient for adaptive algorithms Vh → Vhref .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

2/34 Johann Radon Institute for Computational and Applied Mathematics

Functional type error estimates

For a class of parabolic I-BVP problems ∂tu + L u = f , u(0) = u0, in Ω ⊂ Rd, t ∈ (0, T) u = 0

• n ∂Ω,

with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M(v, D(Ω, u0, f )) ≤ | | |u − v| | | ≤ M(v, D(Ω, u0, f )), minorant error majorant V u v M M universal for any v ∈ V , computable, reliable, i.e., | | |u − v| | | ≤ M(v, D), realistic in comparison to error, i.e., Ieff =

M | | |u−v| | | is close to 1,

consistent, i.e., M(v) is continuous with respect to v and M(u) = 0, efficient for adaptive algorithms Vh → Vhref .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

2/34 Johann Radon Institute for Computational and Applied Mathematics

Functional type error estimates

For a class of parabolic I-BVP problems ∂tu + L u = f , u(0) = u0, in Ω ⊂ Rd, t ∈ (0, T) u = 0

• n ∂Ω,

with u ∈ V is the exact solution (unknown!), v ∈ V is an approximation, D is the problem data, we consider functional a posteriori error estimates M(v, D(Ω, u0, f )) ≤ | | |u − v| | | ≤ M(v, D(Ω, u0, f )), minorant error majorant V u v M M universal for any v ∈ V , computable, reliable, i.e., | | |u − v| | | ≤ M(v, D), realistic in comparison to error, i.e., Ieff =

M | | |u−v| | | is close to 1,

consistent, i.e., M(v) is continuous with respect to v and M(u) = 0, efficient for adaptive algorithms Vh → Vhref .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

3/34 Johann Radon Institute for Computational and Applied Mathematics

Model I-BVP problem

Find u : Q → R satisfying linear parabolic initial-boundary value problem (I-BVP) ∂tu − divx · p = f in Q, p = ∇xu u(x, 0) = u0

• n Σ0,

u = 0

• n Σ,

Ω Σ0 ΣT Ωt [0, T] t x1 x2 x ∈ Ω ⊂ Rd, d = {1, 2, 3}, T > 0 (x, t) ∈ Q := Ω × (0, T) (x, t) ∈ ∂Q := Σ ∪ Σ0 ∪ ΣT Σ := ∂Ω × (0, T) Σ0 := Ω × {0} ΣT := Ω × {T} where ∂t denotes the time derivative, divx and ∇x are divergence and gradient operators in space, respectively, u0 ∈ H1

0(Σ0) is a given initial state,

f is a source function in L2(Q), with uL2(Q) = uQ induced by (v, w)Q =:

• Q v w dxdt.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

3/34 Johann Radon Institute for Computational and Applied Mathematics

Model I-BVP problem

Find u : Q → R satisfying linear parabolic initial-boundary value problem (I-BVP) ∂tu − divx · p = f in Q, p = ∇xu u(x, 0) = u0

• n Σ0,

u = 0

• n Σ,

Ω Σ0 ΣT Ωt [0, T] t x1 x2 x ∈ Ω ⊂ Rd, d = {1, 2, 3}, T > 0 (x, t) ∈ Q := Ω × (0, T) (x, t) ∈ ∂Q := Σ ∪ Σ0 ∪ ΣT Σ := ∂Ω × (0, T) Σ0 := Ω × {0} ΣT := Ω × {T} where ∂t denotes the time derivative, divx and ∇x are divergence and gradient operators in space, respectively, u0 ∈ H1

0(Σ0) is a given initial state,

f is a source function in L2(Q), with uL2(Q) = uQ induced by (v, w)Q =:

• Q v w dxdt.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

3/34 Johann Radon Institute for Computational and Applied Mathematics

Model I-BVP problem

Find u : Q → R satisfying linear parabolic initial-boundary value problem (I-BVP) ∂tu − divx · p = f in Q, p = ∇xu u(x, 0) = u0

• n Σ0,

u = 0

• n Σ,

Ω Σ0 ΣT Ωt [0, T] t x1 x2 x ∈ Ω ⊂ Rd, d = {1, 2, 3}, T > 0 (x, t) ∈ Q := Ω × (0, T) (x, t) ∈ ∂Q := Σ ∪ Σ0 ∪ ΣT Σ := ∂Ω × (0, T) Σ0 := Ω × {0} ΣT := Ω × {T} where ∂t denotes the time derivative, divx and ∇x are divergence and gradient operators in space, respectively, u0 ∈ H1

0(Σ0) is a given initial state,

f is a source function in L2(Q), with uL2(Q) = uQ induced by (v, w)Q =:

• Q v w dxdt.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

3/34 Johann Radon Institute for Computational and Applied Mathematics

Model I-BVP problem

Find u : Q → R satisfying linear parabolic initial-boundary value problem (I-BVP) ∂tu − divx · p = f in Q, p = ∇xu u(x, 0) = u0

• n Σ0,

u = 0

• n Σ,

Ω Σ0 ΣT Ωt [0, T] t x1 x2 x ∈ Ω ⊂ Rd, d = {1, 2, 3}, T > 0 (x, t) ∈ Q := Ω × (0, T) (x, t) ∈ ∂Q := Σ ∪ Σ0 ∪ ΣT Σ := ∂Ω × (0, T) Σ0 := Ω × {0} ΣT := Ω × {T} where ∂t denotes the time derivative, divx and ∇x are divergence and gradient operators in space, respectively, u0 ∈ H1

0(Σ0) is a given initial state,

f is a source function in L2(Q), with uL2(Q) = uQ induced by (v, w)Q =:

• Q v w dxdt.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

4/34 Johann Radon Institute for Computational and Applied Mathematics

Weak formulation

Find a weak solution u ∈ H1,0 (Q) :=

• v ∈ L2(Q) : ∇xv ∈ [L2(Q)]d, v
• Σ = 0
• ,

which satisfies the integral identity for ∀w ∈ H1

0,0(Q) :=

• v ∈ L2(Q) : ∇xv ∈ [L2(Q)]d, ∂tv ∈ L2(Q), v
• Σ = 0, v
• ΣT = 0
• :

a(u, w) = l(w), ∀w ∈ H1

0,0(Q),

with the bilinear form a(u, w) :=

• ∇xu, ∇xw
• Q −
• u, ∂tw
• Q,

and the linear functional l(w) :=

• f , w
• Q + (u0, w)Σ0.

In spirit of

• O. A. Ladyzhenskaya. On solvability of classical boundary value problems for equations of parabolic and hyperbolic
• types. Dokl. Akad. Nauk SSSR, 97(3): 1954, 395–398.
• O. A. Ladyzhenskaya, V. A. Solonnikov, and N.N. Uraltseva. Linear and quasilinear equations of parabolic type.

Nauka, Moscow, 1967. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

5/34 Johann Radon Institute for Computational and Applied Mathematics

Functional a posteriori error analysis for model I-BVP problem

study the distance between generalized solution u ∈ H1

0(Q) of the system

(ut, w)Q + (∇xu, ∇xw)Q = (f , w)Q, ∀w ∈ H1

0(Q),

and any function v ∈ H1

0(Q), which is measured by

| | | u − v | | |2

(ν) := νx,Q ∇x(u − v)2 Q

• energy error

+νt,ΣT u − v 2

ΣT

• error at the last moment

, νx,Q, νt,ΣT > 0.

• S. Repin, Estimates of deviations from exact solutions of I-BVP for the heat equation. Atti Accad. Naz. Lincei Cl. Sci. Fis.
• Mat. Natur. Rend. Lincei, 2002.
• S. Repin and S. K. Tomar, A posteriori error estimates for approximations of evolutionary convection-diffusion problems,

Journal of Mathematical Sciences, 2010.

• P. Neittaanmäki and S. Repin, A posteriori error majorants for approximations of the evolutionary Stokes problem. J.
• Numer. Math, 2010.
• U. Langer, S. Repin, and M. Wolfmayr, Functional a posteriori error estimates for parabolic time-periodic BVPs. CMAM,

2015.

• U. Langer, S. Repin, and M. Wolfmayr, Functional a posteriori error estimates for time-periodic parabolic optimal control

problems, Numer. Func. Anal. Opt., 2016. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

5/34 Johann Radon Institute for Computational and Applied Mathematics

Functional a posteriori error analysis for model I-BVP problem

study the distance between generalized solution u ∈ H1

0(Q) of the system

(ut − vt, w)Q + (∇xu − ∇xv, ∇xw)Q = (f , w)Q − (vt, w)Q − (∇xv, ∇xw)Q, ∀w ∈ H1

0(Q),

and any function v ∈ H1

0(Q), which is measured by

| | | u − v | | |2

(ν) := νx,Q ∇x(u − v)2 Q

• energy error

+νt,ΣT u − v 2

ΣT

• error at the last moment

, νx,Q, νt,ΣT > 0.

• S. Repin, Estimates of deviations from exact solutions of I-BVP for the heat equation. Atti Accad. Naz. Lincei Cl. Sci. Fis.
• Mat. Natur. Rend. Lincei, 2002.
• S. Repin and S. K. Tomar, A posteriori error estimates for approximations of evolutionary convection-diffusion problems,

Journal of Mathematical Sciences, 2010.

• P. Neittaanmäki and S. Repin, A posteriori error majorants for approximations of the evolutionary Stokes problem. J.
• Numer. Math, 2010.
• U. Langer, S. Repin, and M. Wolfmayr, Functional a posteriori error estimates for parabolic time-periodic BVPs. CMAM,

2015.

• U. Langer, S. Repin, and M. Wolfmayr, Functional a posteriori error estimates for time-periodic parabolic optimal control

problems, Numer. Func. Anal. Opt., 2016. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

5/34 Johann Radon Institute for Computational and Applied Mathematics

Functional a posteriori error analysis for model I-BVP problem

study the distance between generalized solution u ∈ H1

0(Q) of the system

(ut −vt, u−v)Q +(∇xu−∇xv, ∇x(u−v))Q = (f , u−v)Q −(vt, u−v)Q −(∇xv, ∇x(u−v))Q, and any function v ∈ H1

0(Q), which is measured by

| | | u − v | | |2

(ν) := νx,Q ∇x(u − v)2 Q

• energy error

+νt,ΣT u − v 2

ΣT

• error at the last moment

, νx,Q, νt,ΣT > 0.

• S. Repin, Estimates of deviations from exact solutions of I-BVP for the heat equation. Atti Accad. Naz. Lincei Cl. Sci. Fis.
• Mat. Natur. Rend. Lincei, 2002.
• S. Repin and S. K. Tomar, A posteriori error estimates for approximations of evolutionary convection-diffusion problems,

Journal of Mathematical Sciences, 2010.

• P. Neittaanmäki and S. Repin, A posteriori error majorants for approximations of the evolutionary Stokes problem. J.
• Numer. Math, 2010.
• U. Langer, S. Repin, and M. Wolfmayr, Functional a posteriori error estimates for parabolic time-periodic BVPs. CMAM,

2015.

• U. Langer, S. Repin, and M. Wolfmayr, Functional a posteriori error estimates for time-periodic parabolic optimal control

problems, Numer. Func. Anal. Opt., 2016. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

6/34 Johann Radon Institute for Computational and Applied Mathematics

Majorants for extended class of I-BVP problem

For ∀v ∈ H1

0(Q) and ∀y ∈ Hdivx ,0(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q)
• :

| | | u − v | | |2

(ν) ≤

inf

y ∈ Hdivx ,0(Q)

M2(v, y), robust to a drastic change in values of reaction λ and convection a in L := −∇x · A∇x + λ + a · ∇x (A = AT > 0, νA|ξ|2 ≤ Aξ · ξ, ξ ∈ Rd , λ − 1

2 divx a ≥ δ ≥ 0, a ∈ [L∞(Ω)]d , λ ∈ L∞(Ω)),

adapted for Ω with complicated geometry + non-trivial BC (mixed ΣD ∪ ΣN), auxiliary y is form wider class ˆ Hdivx ,0(Q) (“broken" fluxes).

• S. M. and S. Repin. Computable estimates of the distance to the exact solution of the evolutionary

reaction-diffusion equation, Appl. Math. and Comput., 2014.

• S. M., P. Neittaanmäki, and S. Repin. A posteriori error estimates for time-dependent reaction-diffusion problems

based on the Payne–Weinberger inequality, Disc. and Cont. Dyn. Sys. - A, 2015.

• S. M. and S. Repin. Estimates of the distance to the exact solution of evolutionary reaction-diffusion problems

based on local Poincaré type inequalities, Zap. Nauchn. Sem. S.-Pb. Otd. Mat. Inst. Steklov, 2014.

• S. M. and S. Repin. Explicit constants in Poincaré-type inequalities for simplicial domains, Computational Methods

in Applied Mathematics, 2016. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

6/34 Johann Radon Institute for Computational and Applied Mathematics

Majorants for extended class of I-BVP problem

For ∀v ∈ H1

0(Q) and ∀y ∈ Hdivx ,0(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q)
• :

| | | u − v | | |2

(ν) ≤

inf

y ∈ Hdivx ,0(Q)

M2(v, y), robust to a drastic change in values of reaction λ and convection a in L := −∇x · A∇x + λ + a · ∇x (A = AT > 0, νA|ξ|2 ≤ Aξ · ξ, ξ ∈ Rd , λ − 1

2 divx a ≥ δ ≥ 0, a ∈ [L∞(Ω)]d , λ ∈ L∞(Ω)),

adapted for Ω with complicated geometry + non-trivial BC (mixed ΣD ∪ ΣN), auxiliary y is form wider class ˆ Hdivx ,0(Q) (“broken" fluxes).

• S. M. and S. Repin. Computable estimates of the distance to the exact solution of the evolutionary

reaction-diffusion equation, Appl. Math. and Comput., 2014.

• S. M., P. Neittaanmäki, and S. Repin. A posteriori error estimates for time-dependent reaction-diffusion problems

based on the Payne–Weinberger inequality, Disc. and Cont. Dyn. Sys. - A, 2015.

• S. M. and S. Repin. Estimates of the distance to the exact solution of evolutionary reaction-diffusion problems

based on local Poincaré type inequalities, Zap. Nauchn. Sem. S.-Pb. Otd. Mat. Inst. Steklov, 2014.

• S. M. and S. Repin. Explicit constants in Poincaré-type inequalities for simplicial domains, Computational Methods

in Applied Mathematics, 2016. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

7/34 Johann Radon Institute for Computational and Applied Mathematics

Majorants for extended class of I-BVP problem

For ∀v ∈ H1

0(Q) and ∀y ∈ Hdivx ,0(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q)
• , we have

| | |u − v| | |2

(γ,δ)≤ M2(v, y) := T

• γ
• µ

√

λ−1/2 divx a rf (v, y)

• 2

Ω

+ α1

C2

FΩ

νA (1 − µ) rf (v, y)2 Ω

+ α2rd(v, y)2

A−1 + α3 (CTr

ΣN )2

νA

rb(v, y)2

A−1

• dt,

where CFΩ is Friedrichs constant and CTr

ΣN is trace constant, and

rf (v, y) = f + divxy − ∂v

∂t − λv − a · ∇xv,

⇐ ∂u ∂t − divxp + λu + a · ∇xu = f , rd(v, y) = y − A∇v, ⇐ p = A∇u, rb(v, y) = y · n − F. ⇐ p · n = F, δ ∈ (0, 2], γ ∈ [ 1

2 , +∞), and µ(x) ∈ [0, 1], 3

• i

1 αi = δ are auxiliary parameters. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

7/34 Johann Radon Institute for Computational and Applied Mathematics

Majorants for extended class of I-BVP problem

For ∀v ∈ H1

0(Q) and ∀y ∈ Hdivx ,0(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q)
• , we have

| | |u − v| | |2

(γ,δ)≤ M2(v, y) := T

• γ
• µ

√

λ−1/2 divx a rf (v, y)

• 2

Ω

+ α1

C2

FΩ

νA (1 − µ) rf (v, y)2 Ω

+ α2rd(v, y)2

A−1 + α3 (CTr

ΣN )2

νA

rb(v, y)2

A−1

• dt,

where CFΩ is Friedrichs constant and CTr

ΣN is trace constant, and

rf (v, y) = f + divxy − ∂v

∂t − λv − a · ∇xv,

⇐ ∂u ∂t − divxp + λu + a · ∇xu = f , rd(v, y) = y − A∇v, ⇐ p = A∇u, rb(v, y) = y · n − F. ⇐ p · n = F, δ ∈ (0, 2], γ ∈ [ 1

2 , +∞), and µ(x) ∈ [0, 1], 3

• i

1 αi = δ are auxiliary parameters. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

7/34 Johann Radon Institute for Computational and Applied Mathematics

Majorants for extended class of I-BVP problem

For ∀v ∈ H1

0(Q) and ∀y ∈ Hdivx ,0(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q)
• , we have

| | |u − v| | |2

(γ,δ)≤ M2(v, y) := T

• γ
• µ

√

λ−1/2 divx a rf (v, y)

• 2

Ω

+ α1

C2

FΩ

νA (1 − µ) rf (v, y)2 Ω

+ α2rd(v, y)2

A−1 + α3 (CTr

ΣN )2

νA

rb(v, y)2

A−1

• dt,

where CFΩ is Friedrichs constant and CTr

ΣN is trace constant, and

rf (v, y) = f + divxy − ∂v

∂t − λv − a · ∇xv,

⇐ ∂u ∂t − divxp + λu + a · ∇xu = f , rd(v, y) = y − A∇v, ⇐ p = A∇u, rb(v, y) = y · n − F. ⇐ p · n = F, δ ∈ (0, 2], γ ∈ [ 1

2 , +∞), and µ(x) ∈ [0, 1], 3

• i

1 αi = δ are auxiliary parameters. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

7/34 Johann Radon Institute for Computational and Applied Mathematics

Majorants for extended class of I-BVP problem

For ∀v ∈ H1

0(Q) and ∀y ∈ Hdivx ,0(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q)
• , we have

| | |u − v| | |2

(γ,δ)≤ M2(v, y) := T

• γ
• µ

√

λ−1/2 divx a rf (v, y)

• 2

Ω

+ α1

C2

FΩ

νA (1 − µ) rf (v, y)2 Ω

+ α2rd(v, y)2

A−1 + α3 (CTr

ΣN )2

νA

rb(v, y)2

A−1

• dt,

where CFΩ is Friedrichs constant and CTr

ΣN is trace constant, and

rf (v, y) = f + divxy − ∂v

∂t − λv − a · ∇xv,

⇐ ∂u ∂t − divxp + λu + a · ∇xu = f , rd(v, y) = y − A∇v, ⇐ p = A∇u, rb(v, y) = y · n − F. ⇐ p · n = F, δ ∈ (0, 2], γ ∈ [ 1

2 , +∞), and µ(x) ∈ [0, 1], 3

• i

1 αi = δ are auxiliary parameters. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

8/34 Johann Radon Institute for Computational and Applied Mathematics

Domain decomposition & local embedding inequalities

subdomains (bounded Lipschitz convex): Ω :=

• Ωi ⊂ OΩ

Ωi, Ωi ∩ Ωj = ∅, i = j, i, j = 1, . . . , N. edges: Γij = Ωi ∩ Ωj, ΓDi = Ωi ∩ ΓD, ΓNi = Ωi ∩ ΓN. local embedding inequalities: Ω ΓD ΓN Ωi Γij ΓDi ΓNi 1 Poincaré inequalities, where the upper bounds of CP

Ωi are derived by Payne, Weinberger (1960):

wΩi ≤ CP

Ωi ∇xwΩi ,

∀w ∈ H1(Ωi) :=

• w ∈ H1(Ωi)
• 1

|Ωi |

• Ωi

w dx = 0

• ,

2 the Poincaré-type inequality for functions with zero mean trace on Γi ⊂ ∂Ωi, where the upper bounds of CTr

Γi

is derived by Nazarov, Repin (2015): wΓi ≤ CTr

Γi ∇xwΩi ,

∀w ∈ H1(Ωi, Γi) :=

• w ∈ H1(Ωi)
• Γi

w dx = 0

• .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

9/34 Johann Radon Institute for Computational and Applied Mathematics

Applications

M2(v, y) =

T

• γ
• µ

√

λ−1/2 divx a rf (v, y)

• 2

Ω

+ α1

C2 FΩ νA (1 − µ) rf (v, y)2 Ω

+ α2rd (v, y)2

A−1 + α3 (CTr ΣN )2 νA

rb(v, y)2

A−1

• dt

1 INCREMENTAL APPROACH: Ω0 v0 = φ(x) t0 Ωtk−1 vk−1 tk−1 Ωtk vk tk Ωtk+1 vk+1 tk+1 T t x1 x2 2 SPACE-TIME APPROACH: t is treated as d + 1 variable.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

9/34 Johann Radon Institute for Computational and Applied Mathematics

Applications

M2(v, y) =

T

• γ
• µ

√

λ−1/2 divx a rf (v, y)

• 2

Ω

+ α1

C2 FΩ νA (1 − µ) rf (v, y)2 Ω

+ α2rd (v, y)2

A−1 + α3 (CTr ΣN )2 νA

rb(v, y)2

A−1

• dt

1 INCREMENTAL APPROACH: Ω0 v0 = φ(x) t0 Ωtk−1 vk−1 tk−1 Ωtk vk tk Ωtk+1 vk+1 tk+1 T t x1 x2 2 SPACE-TIME APPROACH: t is treated as d + 1 variable.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

9/34 Johann Radon Institute for Computational and Applied Mathematics

Applications

M2(v, y) =

T

• γ
• µ

√

λ−1/2 divx a rf (v, y)

• 2

Ω

+ α1

C2 FΩ νA (1 − µ) rf (v, y)2 Ω

+ α2rd (v, y)2

A−1 + α3 (CTr ΣN )2 νA

rb(v, y)2

A−1

• dt

1 INCREMENTAL APPROACH: Ω0 v0 = φ(x) t0 vk−1 Ωtk−1 tk−1 Ωtk vk tk Ωtk+1 vk+1 tk+1 T t x1 x2 2 SPACE-TIME APPROACH: t is treated as d + 1 variable.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

9/34 Johann Radon Institute for Computational and Applied Mathematics

Applications

M2(v, y) =

T

• γ
• µ

√

λ−1/2 divx a rf (v, y)

• 2

Ω

+ α1

C2 FΩ νA (1 − µ) rf (v, y)2 Ω

+ α2rd (v, y)2

A−1 + α3 (CTr ΣN )2 νA

rb(v, y)2

A−1

• dt

1 INCREMENTAL APPROACH: Ω0 v0 = φ(x) t0 vk−1 Ωtk−1 tk−1 Ωtk vk tk Ωtk+1 vk+1 tk+1 T t x1 x2 2 SPACE-TIME APPROACH: t is treated as d + 1 variable.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

9/34 Johann Radon Institute for Computational and Applied Mathematics

Applications

M2(v, y) =

T

• γ
• µ

√

λ−1/2 divx a rf (v, y)

• 2

Ω

+ α1

C2 FΩ νA (1 − µ) rf (v, y)2 Ω

+ α2rd (v, y)2

A−1 + α3 (CTr ΣN )2 νA

rb(v, y)2

A−1

• dt

1 INCREMENTAL APPROACH: Ω0 v0 = φ(x) t0 vk−1 Ωtk−1 tk−1 Ωtk vk tk Ωtk+1 vk+1 tk+1 T t x1 x2 2 SPACE-TIME APPROACH: t is treated as d + 1 variable.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v1

• n

Ω1

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.004 −0.002 0.000 0.002 0.004 0.006 0.008

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v2

• n

Ω2

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.010 −0.005 0.000 0.005 0.010 0.015 0.020

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v3

• n

Ω3

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.02 −0.01 0.00 0.01 0.02 0.03 0.04

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v4

• n

Ω4

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.02 −0.01 0.00 0.01 0.02 0.03 0.04 0.05

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v5

• n

Ω5

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.02 0.00 0.02 0.04 0.06

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v6

• n

Ω6

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.04 −0.02 0.00 0.02 0.04 0.06 0.08

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v7

• n

Ω7

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.04 −0.02 0.00 0.02 0.04 0.06 0.08 0.10

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v8

• n

Ω8

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.04 −0.02 0.00 0.02 0.04 0.06 0.08 0.10

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v9

• n

Ω9

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.04 −0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v10

• n

Ω10

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.06 −0.04 −0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v11

• n

Ω11

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.10 −0.05 0.00 0.05 0.10 0.15

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v12

• n

Ω12

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.10 −0.05 0.00 0.05 0.10 0.15

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v13

• n

Ω13

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.15 −0.10 −0.05 0.00 0.05

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v14

• n

Ω14

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.15 −0.10 −0.05 0.00 0.05

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

10/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Incremental approach

Ω := (−1, 1) × (−1, 1)\[− 1

2 , − 1 2 ] × [0, −1) ⊂ R2, T = 2,

u0 = 0, uD = 0, A = I, f = t sin(t) sin(πx1) + t cos(t) sin(πx2) in Q, vk(x) ∈ P1 and yk(x) ∈ RT1 on Ω × {tk}, y = yk(x) tk+1−t

τ

+ yk+1(x) t−tk

τ

, τ = tk+1 − tk on Qk

T = [tk, tk+1] × Ω,

k = 0, . . . , K − 1, K = 15, Th

• n

Ω v15

• n

Ω15

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(0): 110 EL, 76 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(0): 110 EL, 76 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(1): 160 EL, 133 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(1): 168 EL, 110 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(2): 256 EL, 158 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(2): 259 EL, 160 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(3): 391 EL, 233 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(3): 392 EL, 233 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(4): 602 EL, 347 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(4): 607 EL, 350 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(5): 955 EL, 533 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(5): 951 EL, 531 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(6): 1462 EL, 801 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(6): 1484 EL, 810 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(7): 2250 EL, 1213 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(7): 2277 EL, 1227 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(8): 3458 EL, 1832 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(8): 3519 EL, 1863 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(9): 5202 EL, 2725 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(9): 5304 EL, 2775 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(10): 7865 EL, 4086 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(10): 7977 EL, 4141 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(11): 11790 EL, 6079 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(11): 11970 EL, 6170 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

11/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Comparison of mesh refinement

REFINEMENT REFINEMENT BASED ON THE TRUE ERROR BASED ON THE MAJORANT

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(12): 17305 EL, 8875 ND

−1.0 −0.5 0.0 0.5 1.0 x1 −1.0 −0.5 0.0 0.5 1.0 x2

Q(12): 17825 EL, 9141 ND

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

12/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Reliability and efficiency

[tk, tk+1] | | |e| | |2 M2 Ieff = M

[e]

[t0, t1] 4.23e-06 2.99e-05 2.66 [t1, t2] 2.72e-05 6.34e-05 1.53 [t2, t3] 7.11e-05 1.21e-04 1.31 [t3, t4] 1.32e-04 2.09e-04 1.26 [t4, t5] 2.05e-04 3.06e-04 1.22 [t5, t6] 2.79e-04 4.07e-04 1.21 [t6, t7] 3.49e-04 5.04e-04 1.20 [t7, t8] 4.12e-04 5.93e-04 1.20 [t8, t9] 4.67e-04 6.71e-04 1.20 [t9, t10] 5.13e-04 7.40e-04 1.20 [t10, t11] 5.51e-04 7.99e-04 1.20 [t11, t12] 5.82e-04 8.48e-04 1.21 [t12, t13] 6.07e-04 8.89e-04 1.21 [t13, t14] 6.26e-04 9.22e-04 1.21 [t14, t15] 6.41e-04 9.50e-04 1.22 Efficiency index on Q(k).

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

12/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Reliability and efficiency

[tk, tk+1] | | |e| | |2 M2 Ieff = M

[e]

[t0, t1] 4.23e-06 2.99e-05 2.66 [t1, t2] 2.72e-05 6.34e-05 1.53 [t2, t3] 7.11e-05 1.21e-04 1.31 [t3, t4] 1.32e-04 2.09e-04 1.26 [t4, t5] 2.05e-04 3.06e-04 1.22 [t5, t6] 2.79e-04 4.07e-04 1.21 [t6, t7] 3.49e-04 5.04e-04 1.20 [t7, t8] 4.12e-04 5.93e-04 1.20 [t8, t9] 4.67e-04 6.71e-04 1.20 [t9, t10] 5.13e-04 7.40e-04 1.20 [t10, t11] 5.51e-04 7.99e-04 1.20 [t11, t12] 5.82e-04 8.48e-04 1.21 [t12, t13] 6.07e-04 8.89e-04 1.21 [t13, t14] 6.26e-04 9.22e-04 1.21 [t14, t15] 6.41e-04 9.50e-04 1.22 Efficiency index on Q(k). [t0, t1] = [0.00, 0.13]

20 40 60 80 100 0.5 1 1.5 x 10

−7

number of FEs error 20 40 60 80 100 0.5 1 1.5 x 10

−7

number of FEs majorant

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

12/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Reliability and efficiency

[tk, tk+1] | | |e| | |2 M2 Ieff = M

[e]

[t0, t1] 4.23e-06 2.99e-05 2.66 [t1, t2] 2.72e-05 6.34e-05 1.53 [t2, t3] 7.11e-05 1.21e-04 1.31 [t3, t4] 1.32e-04 2.09e-04 1.26 [t4, t5] 2.05e-04 3.06e-04 1.22 [t5, t6] 2.79e-04 4.07e-04 1.21 [t6, t7] 3.49e-04 5.04e-04 1.20 [t7, t8] 4.12e-04 5.93e-04 1.20 [t8, t9] 4.67e-04 6.71e-04 1.20 [t9, t10] 5.13e-04 7.40e-04 1.20 [t10, t11] 5.51e-04 7.99e-04 1.20 [t11, t12] 5.82e-04 8.48e-04 1.21 [t12, t13] 6.07e-04 8.89e-04 1.21 [t13, t14] 6.26e-04 9.22e-04 1.21 [t14, t15] 6.41e-04 9.50e-04 1.22 Efficiency index on Q(k). [t2, t3] = [0.27, 0.40]

50 100 150 200 250 0.2 0.4 0.6 0.8 1 1.2 x 10

−6

number of FEs error 50 100 150 200 250 0.2 0.4 0.6 0.8 1 1.2 x 10

−6

number of FEs majorant

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

12/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Reliability and efficiency

[tk, tk+1] | | |e| | |2 M2 Ieff = M

[e]

[t0, t1] 4.23e-06 2.99e-05 2.66 [t1, t2] 2.72e-05 6.34e-05 1.53 [t2, t3] 7.11e-05 1.21e-04 1.31 [t3, t4] 1.32e-04 2.09e-04 1.26 [t4, t5] 2.05e-04 3.06e-04 1.22 [t5, t6] 2.79e-04 4.07e-04 1.21 [t6, t7] 3.49e-04 5.04e-04 1.20 [t7, t8] 4.12e-04 5.93e-04 1.20 [t8, t9] 4.67e-04 6.71e-04 1.20 [t9, t10] 5.13e-04 7.40e-04 1.20 [t10, t11] 5.51e-04 7.99e-04 1.20 [t11, t12] 5.82e-04 8.48e-04 1.21 [t12, t13] 6.07e-04 8.89e-04 1.21 [t13, t14] 6.26e-04 9.22e-04 1.21 [t14, t15] 6.41e-04 9.50e-04 1.22 Efficiency index on Q(k). [t4, t5] = [0.53, 0.67]

100 200 300 400 500 600 0.2 0.4 0.6 0.8 1 x 10

−6

number of FEs error 100 200 300 400 500 600 0.2 0.4 0.6 0.8 1 x 10

−6

number of FEs majorant

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

12/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Reliability and efficiency

[tk, tk+1] | | |e| | |2 M2 Ieff = M

[e]

[t0, t1] 4.23e-06 2.99e-05 2.66 [t1, t2] 2.72e-05 6.34e-05 1.53 [t2, t3] 7.11e-05 1.21e-04 1.31 [t3, t4] 1.32e-04 2.09e-04 1.26 [t4, t5] 2.05e-04 3.06e-04 1.22 [t5, t6] 2.79e-04 4.07e-04 1.21 [t6, t7] 3.49e-04 5.04e-04 1.20 [t7, t8] 4.12e-04 5.93e-04 1.20 [t8, t9] 4.67e-04 6.71e-04 1.20 [t9, t10] 5.13e-04 7.40e-04 1.20 [t10, t11] 5.51e-04 7.99e-04 1.20 [t11, t12] 5.82e-04 8.48e-04 1.21 [t12, t13] 6.07e-04 8.89e-04 1.21 [t13, t14] 6.26e-04 9.22e-04 1.21 [t14, t15] 6.41e-04 9.50e-04 1.22 Efficiency index on Q(k). [t6, t7] = [0.80, 0.93]

200 400 600 800 1000 1200 1400 1 2 3 x 10

−7

number of FEs error 200 400 600 800 1000 1200 1400 1 2 3 x 10

−7

number of FEs majorant

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

12/34 Johann Radon Institute for Computational and Applied Mathematics

Example 1. Reliability and efficiency

[tk, tk+1] | | |e| | |2 M2 Ieff = M

[e]

[t0, t1] 4.23e-06 2.99e-05 2.66 [t1, t2] 2.72e-05 6.34e-05 1.53 [t2, t3] 7.11e-05 1.21e-04 1.31 [t3, t4] 1.32e-04 2.09e-04 1.26 [t4, t5] 2.05e-04 3.06e-04 1.22 [t5, t6] 2.79e-04 4.07e-04 1.21 [t6, t7] 3.49e-04 5.04e-04 1.20 [t7, t8] 4.12e-04 5.93e-04 1.20 [t8, t9] 4.67e-04 6.71e-04 1.20 [t9, t10] 5.13e-04 7.40e-04 1.20 [t10, t11] 5.51e-04 7.99e-04 1.20 [t11, t12] 5.82e-04 8.48e-04 1.21 [t12, t13] 6.07e-04 8.89e-04 1.21 [t13, t14] 6.26e-04 9.22e-04 1.21 [t14, t15] 6.41e-04 9.50e-04 1.22 Efficiency index on Q(k). [t8, t9] = [1.07, 1.20]

500 1000 1500 2000 2500 3000 3500 0.5 1 1.5 2 x 10

−7

number of FEs error 500 1000 1500 2000 2500 3000 3500 0.5 1 1.5 2 x 10

−7

number of FEs majorant

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

13/34 Johann Radon Institute for Computational and Applied Mathematics

Example 2. Space-time approach

(0, 1) ⊂ R1, T = 1, homogeneous Dirichlet BC on ΣD, f = x (1 − x) (2t + 1) + 2

• t2 + t + 1
• (x (1 − x)) in Q,

u = x (1 − x)

• t2 + t + 1
• ,

v ∈ P1 and y ∈ P2 in Q, initial mesh T3×3.

x 0.00 0.33 0.67 t 0.00 0.33 0.67 u

• 0.10

0.20 0.50

Approximate solution v on Θ17×17.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

14/34 Johann Radon Institute for Computational and Applied Mathematics

Example 2. Mesh uniform refinement

10

−3

10

−2

10

−1

10 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 log h log M

2, log [e]2

h2 M

2

[e]2

Optimal convergence of total error and majorant. REF EL [e] 2 M2 Ieff 1 8 3.52e-01 4.08e-01 1.08 2 32 9.11e-02 1.06e-01 1.08 3 128 2.30e-02 2.72e-02 1.09 4 512 5.75e-03 6.85e-03 1.09 5 2048 1.43e-03 1.72e-03 1.09 6 8192 3.59e-04 4.30e-04 1.09 7 131072 2.24e-05 2.69e-05 1.09 8 524288 5.62e-06 6.74e-06 1.10 9 2097152 1.40e-06 1.68e-06 1.10 Total error, majorant, and corresponding efficiency index.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

15/34 Johann Radon Institute for Computational and Applied Mathematics

Example 2. | | |e| | |2 and M

2 distribution on REF # = 3, 4 20 40 60 80 100 120 2 4 6 8 x 10

−4

e2

d, m2 d

e2

d

m2

d

REF 3: 128 EL, 81 ND e 2

d = 2.2969e-02, m2 d = 2.2999e-02

100 200 300 400 500 1 2 3 4 5 x 10

−5

e2

d, m2 d

e2

d

m2

d

REF 4: 512 EL, 289 ND e 2

d = 1.4393e-03, m2 d = 1.4395e-03 Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

16/34 Johann Radon Institute for Computational and Applied Mathematics

Main idea on IgA framework

Physical domain Q ⊂ Rd+1, is defined from Parametric domain Q := (0, 1)d+1 by the Geometrical mapping Φ : Q → Q = Φ( Q) ⊂ Rd+1, Φ(ξ) =

i∈I

Bi,p(ξ)Pi, where Bi,p, i ∈ I are the NURBS (BSplines) basis function, {Pi}i∈I ∈ Rd+1 are the control points.

Kh is a mesh defined on Q: ˆ h := max

• K∈

Kh

{ˆ h

K},

ˆ h

K := diam(

K).

• Kh is a mesh defined on Q:

Kh :=

• K = Φ(

K) : K ∈ Kh

• ,

a b Σ0 ΣT T Q Σ x t ˆ x ˆ t ˆ Q Φ−1 Φ 1 1 where h := max

K∈Kh

{ hKi }, hK := ∇ΦL∞(K)ˆ h

K. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

17/34 Johann Radon Institute for Computational and Applied Mathematics

Example of 1d parametrical domain

Let Q := (0, 1) and Kh is the mesh defined on it:

• 1. Let p ≥ 2 be a polynomial degree, n denote the number of basis function.
• 2. Knot vector in 1d is a non-decreasing set of coordinates in the parameter domain:

Ξ = {ξ1, . . . , ξn+p+1}, ξi ∈ R, ξ1 = 0 and ξn+p+1 = 1. Basis functions with p = 2 for the open knot vector Ξ = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5}. Basis functions are Bi,0 : (0, 1) → R are defined by Cox-de Boor recursion:

• Bi,p :=

ξ−ξi ξi+p−ξi

• Bi,p−1 +

ξi+p+1−ξ ξi+p+1−ξi+1

• Bi+1,p−1,
• Bi,0:=
• 1

if ξi ≤ ξ ≤ ξi+1,

• therwise.
• 3. Knots can be repeated, such that mi is the multiplicity i-th knot.

T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs Isogeometric Analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement., CMAME 2005.

• Y. Bazilevs, L. Beirao da Veiga, J. A. Cottrell, T. J. R. Hughes, and G. Sangalli. Isogeometric analysis:

approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci., 2006. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

17/34 Johann Radon Institute for Computational and Applied Mathematics

Example of 1d parametrical domain

Let Q := (0, 1) and Kh is the mesh defined on it:

• 1. Let p ≥ 2 be a polynomial degree, n denote the number of basis function.
• 2. Knot vector in 1d is a non-decreasing set of coordinates in the parameter domain:

Ξ = {ξ1, . . . , ξn+p+1}, ξi ∈ R, ξ1 = 0 and ξn+p+1 = 1. Basis functions with p = 2 for the open knot vector Ξ = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5}. Basis functions are Bi,0 : (0, 1) → R are defined by Cox-de Boor recursion:

• Bi,p :=

ξ−ξi ξi+p−ξi

• Bi,p−1 +

ξi+p+1−ξ ξi+p+1−ξi+1

• Bi+1,p−1,
• Bi,0:=
• 1

if ξi ≤ ξ ≤ ξi+1,

• therwise.
• 3. Knots can be repeated, such that mi is the multiplicity i-th knot.

T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs Isogeometric Analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement., CMAME 2005.

• Y. Bazilevs, L. Beirao da Veiga, J. A. Cottrell, T. J. R. Hughes, and G. Sangalli. Isogeometric analysis:

approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci., 2006. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

17/34 Johann Radon Institute for Computational and Applied Mathematics

Example of 1d parametrical domain

Let Q := (0, 1) and Kh is the mesh defined on it:

• 1. Let p ≥ 2 be a polynomial degree, n denote the number of basis function.
• 2. Knot vector in 1d is a non-decreasing set of coordinates in the parameter domain:

Ξ = {ξ1, . . . , ξn+p+1}, ξi ∈ R, ξ1 = 0 and ξn+p+1 = 1. Basis functions with p = 2 for the open knot vector Ξ = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5}. Basis functions are Bi,0 : (0, 1) → R are defined by Cox-de Boor recursion:

• Bi,p :=

ξ−ξi ξi+p−ξi

• Bi,p−1 +

ξi+p+1−ξ ξi+p+1−ξi+1

• Bi+1,p−1,
• Bi,0:=
• 1

if ξi ≤ ξ ≤ ξi+1,

• therwise.
• 3. Knots can be repeated, such that mi is the multiplicity i-th knot.

T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs Isogeometric Analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement., CMAME 2005.

• Y. Bazilevs, L. Beirao da Veiga, J. A. Cottrell, T. J. R. Hughes, and G. Sangalli. Isogeometric analysis:

approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci., 2006. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

17/34 Johann Radon Institute for Computational and Applied Mathematics

Example of 1d parametrical domain

Let Q := (0, 1) and Kh is the mesh defined on it:

• 1. Let p ≥ 2 be a polynomial degree, n denote the number of basis function.
• 2. Knot vector in 1d is a non-decreasing set of coordinates in the parameter domain:

Ξ = {ξ1, . . . , ξn+p+1}, ξi ∈ R, ξ1 = 0 and ξn+p+1 = 1. Basis functions with p = 2 for the open knot vector Ξ = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5}. Basis functions are Bi,0 : (0, 1) → R are defined by Cox-de Boor recursion:

• Bi,p :=

ξ−ξi ξi+p−ξi

• Bi,p−1 +

ξi+p+1−ξ ξi+p+1−ξi+1

• Bi+1,p−1,
• Bi,0:=
• 1

if ξi ≤ ξ ≤ ξi+1,

• therwise.
• 3. Knots can be repeated, such that mi is the multiplicity i-th knot.

T.J.R. Hughes, J.A. Cottrell and Y. Bazilevs Isogeometric Analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement., CMAME 2005.

• Y. Bazilevs, L. Beirao da Veiga, J. A. Cottrell, T. J. R. Hughes, and G. Sangalli. Isogeometric analysis:

approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci., 2006. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

18/34 Johann Radon Institute for Computational and Applied Mathematics

Main properties of BSplines (NURBS)

Basis functions with p = 2 for the open knot vector Ξ = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5}. Basis functions are (p − mi) times differentiable across a i-th knot of multiplicity mi:

• Bi,p ∈ Cp−mi .

On the boundary of the patch, he multiplicity is m1 = mn+p+1 = p + 1:

• B1,p,

Bn+p+1,p ∈ C−1. Advantages for the a posteriori error estimates: Polynomial degree of BSplines p ≥ 2 & multiplicity ≤ p − 1 of the inner knots ⇒ at least C1-continuous approximations in the inner knots of the patch. y = ∇xv is automatically in H(Ω, div).

• S. K. Kleiss and S. K. Tomar, Guaranteed and sharp a posteriori error estimates in isogeometric analysis,

Computers & Mathematics with Applications 70 (3), 167-190, 2015. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

18/34 Johann Radon Institute for Computational and Applied Mathematics

Main properties of BSplines (NURBS)

Basis functions with p = 2 for the open knot vector Ξ = {0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5}. Basis functions are (p − mi) times differentiable across a i-th knot of multiplicity mi:

• Bi,p ∈ Cp−mi .

On the boundary of the patch, he multiplicity is m1 = mn+p+1 = p + 1:

• B1,p,

Bn+p+1,p ∈ C−1. Advantages for the a posteriori error estimates: Polynomial degree of BSplines p ≥ 2 & multiplicity ≤ p − 1 of the inner knots ⇒ at least C1-continuous approximations in the inner knots of the patch. y = ∇xv is automatically in H(Ω, div).

• S. K. Kleiss and S. K. Tomar, Guaranteed and sharp a posteriori error estimates in isogeometric analysis,

Computers & Mathematics with Applications 70 (3), 167-190, 2015. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

19/34 Johann Radon Institute for Computational and Applied Mathematics

Discretization spaces for the space-time IgA scheme

The discretization spaces on Q is constructed by the B-splines(NURBS) basis functions: uh ∈ Vh := span

• φh,i =

Bi,p ◦ Φ−1

i∈I:

uh(x, t) =

• i∈I

ui φh,i(x, t), uh := [ui]i∈I ∈ RNh=|I|. Moreover, V0h := Vh ∩ H1

0,0(Q),

and V0h,∗ := V 2,1 (Q) + V0h, where H1

0,0(Q) :=

• v ∈ H1 : v|Σ0∪Σ = 0
• and

V s

0,0 := Hs(Q) ∩ H1 0,0(Q). Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

20/34 Johann Radon Institute for Computational and Applied Mathematics

Space-time IgA discrete scheme

Testing parabolic I-BVP by a function with a time-upwind w = λ vh + µ ∂tvh, where λ = 1 and µ = δh, i.e., vh + δh ∂tvh, δh = θh, θ > 0, vh ∈ V0h ⊂ H1

0,0(Q),

we arrive at the following space-time IgA discrete formulation: Find uh ∈ V0h ⊂ H1

0,0(Q) satisfying

ah(uh, vh) = l(vh), ∀vh ∈ V0h, where ah(uh, vh) :=

• ∂tuh, vh + δh ∂tvh
• Q +
• ∇xuh, ∇x(vh + δh ∂tvh)
• Q

and lh(vh) := (f , vh + δh vh)Q.

• U. Langer, S. E. Moore, M. Neumüller, Space-Time Isogeometric Analysis of Parabolic Evolution Equations.
• Comput. Methods Appl. Mech. Engrg., 2016.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

21/34 Johann Radon Institute for Computational and Applied Mathematics

Properties of the space-time IgA scheme

• 1. The form ah(uh, vh) : V0h × V0h is V0h- coervice w.r.t. to the norm

| | |vh| | |2

h := ∇xvh2 Q + δh ∂tvh2 Q + vh2 ΣT + δh ∇xvh2 ΣT .

• 2. The form ah(·, ·) is uniformly bounded on V0h,∗ × V0h, where V0h,∗ := V 2,1

+ V0h is equipped with norm | | |v| | |2

h,∗ := |

| |v| | |2

h + δ−1 h

v2

Q,

∀v ∈ V0h,∗.

• 3. For 2 ≤ s ≤ p + 1 and v ∈ V s

0,0, there exists a projective operator Πh : V s 0,0 → V0h

satisfying a priori error estimates: | | |v − Πhv| | |h ≤ C1 hs−1vHs(Q) and | | |v − Πhv| | |h,∗≤ C2 hs−1vHs(Q), C1, C2 > 0.

• 4. The form ah(·, ·) is consistent.
• 5. For the exact solution u ∈ V s

0 , s ≥ 2 and approximation uh ∈ V0h, there exists a

discretization error estimate u − uhh ≤ C hr−1 uHr (Q), C > 0 and r = min{s, p + 1}.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

21/34 Johann Radon Institute for Computational and Applied Mathematics

Properties of the space-time IgA scheme

• 1. The form ah(uh, vh) : V0h × V0h is V0h- coervice w.r.t. to the norm

| | |vh| | |2

h := ∇xvh2 Q + δh ∂tvh2 Q + vh2 ΣT + δh ∇xvh2 ΣT .

• 2. The form ah(·, ·) is uniformly bounded on V0h,∗ × V0h, where V0h,∗ := V 2,1

+ V0h is equipped with norm | | |v| | |2

h,∗ := |

| |v| | |2

h + δ−1 h

v2

Q,

∀v ∈ V0h,∗.

• 3. For 2 ≤ s ≤ p + 1 and v ∈ V s

0,0, there exists a projective operator Πh : V s 0,0 → V0h

satisfying a priori error estimates: | | |v − Πhv| | |h ≤ C1 hs−1vHs(Q) and | | |v − Πhv| | |h,∗≤ C2 hs−1vHs(Q), C1, C2 > 0.

• 4. The form ah(·, ·) is consistent.
• 5. For the exact solution u ∈ V s

0 , s ≥ 2 and approximation uh ∈ V0h, there exists a

discretization error estimate u − uhh ≤ C hr−1 uHr (Q), C > 0 and r = min{s, p + 1}.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

21/34 Johann Radon Institute for Computational and Applied Mathematics

Properties of the space-time IgA scheme

• 1. The form ah(uh, vh) : V0h × V0h is V0h- coervice w.r.t. to the norm

| | |vh| | |2

h := ∇xvh2 Q + δh ∂tvh2 Q + vh2 ΣT + δh ∇xvh2 ΣT .

• 2. The form ah(·, ·) is uniformly bounded on V0h,∗ × V0h, where V0h,∗ := V 2,1

+ V0h is equipped with norm | | |v| | |2

h,∗ := |

| |v| | |2

h + δ−1 h

v2

Q,

∀v ∈ V0h,∗.

• 3. For 2 ≤ s ≤ p + 1 and v ∈ V s

0,0, there exists a projective operator Πh : V s 0,0 → V0h

satisfying a priori error estimates: | | |v − Πhv| | |h ≤ C1 hs−1vHs(Q) and | | |v − Πhv| | |h,∗≤ C2 hs−1vHs(Q), C1, C2 > 0.

• 4. The form ah(·, ·) is consistent.
• 5. For the exact solution u ∈ V s

0 , s ≥ 2 and approximation uh ∈ V0h, there exists a

discretization error estimate u − uhh ≤ C hr−1 uHr (Q), C > 0 and r = min{s, p + 1}.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

21/34 Johann Radon Institute for Computational and Applied Mathematics

Properties of the space-time IgA scheme

• 1. The form ah(uh, vh) : V0h × V0h is V0h- coervice w.r.t. to the norm

| | |vh| | |2

h := ∇xvh2 Q + δh ∂tvh2 Q + vh2 ΣT + δh ∇xvh2 ΣT .

• 2. The form ah(·, ·) is uniformly bounded on V0h,∗ × V0h, where V0h,∗ := V 2,1

+ V0h is equipped with norm | | |v| | |2

h,∗ := |

| |v| | |2

h + δ−1 h

v2

Q,

∀v ∈ V0h,∗.

• 3. For 2 ≤ s ≤ p + 1 and v ∈ V s

0,0, there exists a projective operator Πh : V s 0,0 → V0h

satisfying a priori error estimates: | | |v − Πhv| | |h ≤ C1 hs−1vHs(Q) and | | |v − Πhv| | |h,∗≤ C2 hs−1vHs(Q), C1, C2 > 0.

• 4. The form ah(·, ·) is consistent.
• 5. For the exact solution u ∈ V s

0 , s ≥ 2 and approximation uh ∈ V0h, there exists a

discretization error estimate u − uhh ≤ C hr−1 uHr (Q), C > 0 and r = min{s, p + 1}.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

21/34 Johann Radon Institute for Computational and Applied Mathematics

Properties of the space-time IgA scheme

• 1. The form ah(uh, vh) : V0h × V0h is V0h- coervice w.r.t. to the norm

| | |vh| | |2

h := ∇xvh2 Q + δh ∂tvh2 Q + vh2 ΣT + δh ∇xvh2 ΣT .

• 2. The form ah(·, ·) is uniformly bounded on V0h,∗ × V0h, where V0h,∗ := V 2,1

+ V0h is equipped with norm | | |v| | |2

h,∗ := |

| |v| | |2

h + δ−1 h

v2

Q,

∀v ∈ V0h,∗.

• 3. For 2 ≤ s ≤ p + 1 and v ∈ V s

0,0, there exists a projective operator Πh : V s 0,0 → V0h

satisfying a priori error estimates: | | |v − Πhv| | |h ≤ C1 hs−1vHs(Q) and | | |v − Πhv| | |h,∗≤ C2 hs−1vHs(Q), C1, C2 > 0.

• 4. The form ah(·, ·) is consistent.
• 5. For the exact solution u ∈ V s

0 , s ≥ 2 and approximation uh ∈ V0h, there exists a

discretization error estimate u − uhh ≤ C hr−1 uHr (Q), C > 0 and r = min{s, p + 1}.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

22/34 Johann Radon Institute for Computational and Applied Mathematics

Weak formulation of stabilized parabolic I-BVP

Consider alternative test function λ w + µ ∂tw, ∀w ∈ H1

0(Q) and λ, µ ≥ 0:

a′(u, λ w + µ ∂tw) := (ut, λ w + µ ∂tw)Q + (∇xu, ∇x(λ w + µ ∂tw))Q = (f , λ w + µ ∂tw)Q =: l′(u, λ w + µ ∂tw). For ∀v ∈ H1

0(Q), the error between u and v is measured in terms of norm

u − v 2

(ν) := νx,Q ∇x(u − v)2 Q + νt,Q ∂t(u − v)2 Q

• energy error

+ νx,ΣT ∇x(u − v) 2

ΣT + νt,ΣT u − v 2 ΣT

• error at the last moment

, νx,Q, νt,Q, νt,ΣT , νt,ΣT > 0.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

23/34 Johann Radon Institute for Computational and Applied Mathematics

Derivation of advanced form of majorant

• 1. Regularity assumption on u, i.e.,

H∆x ,1 (Q) :=

• v ∈ H1

0(Q) : ∆xv ∈ L2(Q)

• r H∂t∇x ,1

(Q) :=

• v ∈ H1

0(Q) : ∂t(∇xv) ∈ L2(Q)

• 2. Introduce Hilbert spaces for auxiliary vector-valued functions y, i.e.,

Hdiv,0(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q)
• r

Hdiv,1(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q), ∂ty ∈ [L2(Q)]d

, which satisfies the identity (divxy, λ e + µ ∂te)Q + (y, ∇x(λ e + µ ∂te))Q = 0.

• 3. Two forms of the majorants are obtained from

λ ∇xe2

Q + µ ∂t e2 Q + 1 2 (µ ∇xe2 ΣT + λe2 ΣT )

= λ

• (f + divxy − ∂tv, e)Q + (y − ∇xv, ∇xe)Q
• + µ
• (f + divxy − ∂tv, ∂te)Q + (y − ∇xv, ∇x∂te)Q
• .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

23/34 Johann Radon Institute for Computational and Applied Mathematics

Derivation of advanced form of majorant

• 1. Regularity assumption on u, i.e.,

H∆x ,1 (Q) :=

• v ∈ H1

0(Q) : ∆xv ∈ L2(Q)

• r H∂t∇x ,1

(Q) :=

• v ∈ H1

0(Q) : ∂t(∇xv) ∈ L2(Q)

• 2. Introduce Hilbert spaces for auxiliary vector-valued functions y, i.e.,

Hdiv,0(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q)
• r

Hdiv,1(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q), ∂ty ∈ [L2(Q)]d

, which satisfies the identity (divxy, λ e + µ ∂te)Q + (y, ∇x(λ e + µ ∂te))Q = 0.

• 3. Two forms of the majorants are obtained from

λ ∇xe2

Q + µ ∂t e2 Q + 1 2 (µ ∇xe2 ΣT + λe2 ΣT )

= λ

• (f + divxy − ∂tv, e)Q + (y − ∇xv, ∇xe)Q
• + µ
• (f + divxy − ∂tv, ∂te)Q + (y − ∇xv, ∇x∂te)Q
• .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

23/34 Johann Radon Institute for Computational and Applied Mathematics

Derivation of advanced form of majorant

• 1. Regularity assumption on u, i.e.,

H∆x ,1 (Q) :=

• v ∈ H1

0(Q) : ∆xv ∈ L2(Q)

• r H∂t∇x ,1

(Q) :=

• v ∈ H1

0(Q) : ∂t(∇xv) ∈ L2(Q)

• 2. Introduce Hilbert spaces for auxiliary vector-valued functions y, i.e.,

Hdiv,0(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q)
• r

Hdiv,1(Q) :=

• y ∈ [L2(Q)]d : divxy ∈ L2(Q), ∂ty ∈ [L2(Q)]d

, which satisfies the identity (divxy, λ e + µ ∂te)Q + (y, ∇x(λ e + µ ∂te))Q = 0.

• 3. Two forms of the majorants are obtained from

λ ∇xe2

Q + µ ∂t e2 Q + 1 2 (µ ∇xe2 ΣT + λe2 ΣT )

= λ

• (f + divxy − ∂tv, e)Q + (y − ∇xv, ∇xe)Q
• + µ
• (f + divxy − ∂tv, ∂te)Q + (y − ∇xv, ∇x∂te)Q
• .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

24/34 Johann Radon Institute for Computational and Applied Mathematics

1st form of the majorant

Theorem For ∀v ∈ H∆x ,1 (Q) ∩ H∂t∇x ,1 (Q) and ∀y ∈ Hdiv,0(Q), error can be estimated as follows: (2 − 1

γ )(λ ∇xe2 Q + µ ∂te2 Q) + µ ∇xe2 ΣT + λe2 ΣT =: |

| |e| | |2

(γ,µ,λ)

≤ MI(v, y; γ, αi) := γ

• λ
• (1 + α1) rd2

Q + (1 + 1 α1 ) C2 FΩ req2 Q

• + µ
• (1 + α2) divxrd2

Q + (1 + 1 α2 ) req2 Q

• ,

where req(v, y) = f + divxy − ∂tv, ⇐ ∂tu − divxp = f rd(v, y) = y − ∇xv, ⇐ p = ∇xu λ, µ > 0, and γ ∈ 1

2 , +∞) and αi, i = 1, 2 > 0 are auxiliary parameters. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

24/34 Johann Radon Institute for Computational and Applied Mathematics

1st form of the majorant

Theorem For ∀v ∈ H∆x ,1 (Q) ∩ H∂t∇x ,1 (Q) and ∀y ∈ Hdiv,0(Q), error can be estimated as follows: (2 − 1

γ )(λ ∇xe2 Q + µ ∂te2 Q) + µ ∇xe2 ΣT + λe2 ΣT =: |

| |e| | |2

(γ,µ,λ)

≤ MI(v, y; γ, αi) := γ

• λ
• (1 + α1) rd2

Q + (1 + 1 α1 ) C2 FΩ req2 Q

• + µ
• (1 + α2) divxrd2

Q + (1 + 1 α2 ) req2 Q

• ,

where req(v, y) = f + divxy − ∂tv, ⇐ ∂tu − divxp = f rd(v, y) = y − ∇xv, ⇐ p = ∇xu λ, µ > 0, and γ ∈ 1

2 , +∞) and αi, i = 1, 2 > 0 are auxiliary parameters. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

25/34 Johann Radon Institute for Computational and Applied Mathematics

1st form of majorant for space-time IgA scheme (λ = 1 and µ = δh)

Corollary 1 (Follows from Theorem on 1st form of generilized majorant) For ∀v ∈ H∆x ,1 (Q) ∩ H∂t∇x ,1 (Q) and y ∈ Hdiv,0(Q), the following estimate of the error is presented as follows: (2 − 1

γ )( ∇xe2 Q + δh ∂te2 Q) + δh ∇xe2 ΣT + e2 ΣT ≤ MI h(v, y; γ, αi)

:= γ

• (1 + α1) rd2

Q + (1 + 1 α1 ) C2 FΩ req2 Q

+ δh

• (1 + α2) divxrd2

Q + (1 + 1 α2 ) req2 Q

• ,

where rd and req are residuals of two basic equations, δh = θh, θ > 0, is a parameter of the scheme, and γ ∈ 1

2 , +∞), αi, i = 1, 2 > 0 are positive parameters. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

26/34 Johann Radon Institute for Computational and Applied Mathematics

2nd form of the majorant

Theorem For ∀v ∈ H∂t∇x ,1 (Q) and ∀y ∈ Hdiv,1(Q), the estimate of the e = u − v ∈ H1

0(Q) ∩ V 2,1

(Q) of the second form has the following form: (2 − 1

ζ )(λ ∇xe2 Q + µ ∂te2 Q) + µ (1 − 1 ǫ )∇xe2 ΣT + λ e2 ΣT =: |

| |e| | |2

(ǫ,ζ,µ,λ)

≤ MII(v, y; ζ, βi, ǫ) := µ(ǫrd2

ΣT + ζ req2 Q)

+ ζ

• λ2

(1 + β1)

• (1 + β2) rd2

Q + (1 + 1 β2 )C2 FΩ req2 Q

• + µ2 (1 +

1 β1 ) ∂trd2

• ,

where req(v, y) = f + divxy − ∂tv, ⇐ ∂tu − divxp = f rd(v, y) = y − ∇xv, ⇐ p = ∇xu λ, µ > 0, and ζ ∈ 1

2 , +∞), ǫ ∈ [1, +∞), and βi, i = 1, 2 are positive parameters. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

27/34 Johann Radon Institute for Computational and Applied Mathematics

2nd form of majorant for space-time IgA scheme (λ = 1 and µ = δh)

Corollary 2 (Follows from Theorem on 2nd form of generilized majorant) For ∀v ∈ H∂t∇x ,1 (Q) and y ∈ Hdiv,1(Q), the following estimate of the error is presented as follows: (2 − 1

ζ )( ∇xe2 Q + δh ∂te2 Q) + δh (1 − 1 ǫ )∇xe2 ΣT + e2 ΣT ≤ MII h (v, y; ζ, βi, ǫ)

:= ǫ δh rd2

ΣT + δh ζ req2 Q

+ ζ

• (1 + β1)
• (1 + β2) rd2

Q + (1 + 1 β2 )C2 FΩ req2 Q

• + (1 +

1 β1 )δ2 h ∂trd2 Q

• ,

where rd and req are residuals of two basic equations, δh = θh, θ > 0, is a parameter of the scheme, and ζ ∈ 1

2 , +∞), ǫ ∈ [1, +∞), and βi, i = 1, 2 are positive parameters. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

28/34 Johann Radon Institute for Computational and Applied Mathematics

Conclusions

For for non-moving spatial domains: 1 Efficient space-time IgA method for parabolic I-BVP (due to the ellipticity of the discrete scheme). 2 General estimates that provide error control for collection of norms and can be combined with space-time methods presented in:

• U. Langer, S. E. Moore, M. Neumüller, Space-Time Isogeometric Analysis of Parabolic Evolution
• Equations. Comput. Methods Appl. Mech. Engrg., 306:342–363, 2016.
• O. Steinbach, Space-Time Finite Element Methods for Parabolic Problems, 15(4): 2015, pp. 551–566.

...

Further work: Error estimates for 1 general moving spatial domains, 2 the schemes allowing discontinuity in time, 3 for the domains with complex geometries presented as a

collection of non-overlapping multipatches.

Localized discrete space-time IgA scheme, where h-dependent parameter is replaced by hK-dependent, where K is the element of triangulation Th.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

28/34 Johann Radon Institute for Computational and Applied Mathematics

Conclusions

For for growing spatial domains: 1 Efficient space-time IgA method for parabolic I-BVP (due to the ellipticity of the discrete scheme). 2 General estimates that provide error control for collection of norms and can be combined with space-time methods presented in:

• U. Langer, S. E. Moore, M. Neumüller, Space-Time Isogeometric Analysis of Parabolic Evolution
• Equations. Comput. Methods Appl. Mech. Engrg., 306:342–363, 2016.
• O. Steinbach, Space-Time Finite Element Methods for Parabolic Problems, 15(4): 2015, pp. 551–566.

...

Further work: Error estimates for 1 general moving spatial domains, 2 the schemes allowing discontinuity in time, 3 for the domains with complex geometries presented as a

collection of non-overlapping multipatches.

Localized discrete space-time IgA scheme, where h-dependent parameter is replaced by hK-dependent, where K is the element of triangulation Th.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

28/34 Johann Radon Institute for Computational and Applied Mathematics

Conclusions

For for general moving spatial domains: 1 Efficient space-time IgA method for parabolic I-BVP (due to the ellipticity of the discrete scheme). 2 General estimates (?) that provide error control for collection of norms and can be combined with space-time methods presented in:

• U. Langer, S. E. Moore, M. Neumüller, Space-Time Isogeometric Analysis of Parabolic Evolution
• Equations. Comput. Methods Appl. Mech. Engrg., 306:342–363, 2016.
• O. Steinbach, Space-Time Finite Element Methods for Parabolic Problems, 15(4): 2015, pp. 551–566.

...

Further work: Error estimates for 1 general moving spatial domains, 2 the schemes allowing discontinuity in time, 3 for the domains with complex geometries presented as a

collection of non-overlapping multipatches.

Localized discrete space-time IgA scheme, where h-dependent parameter is replaced by hK-dependent, where K is the element of triangulation Th.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

28/34 Johann Radon Institute for Computational and Applied Mathematics

Conclusions

For for general moving spatial domains: 1 Efficient space-time IgA method for parabolic I-BVP (due to the ellipticity of the discrete scheme). 2 General estimates (?) that provide error control for collection of norms and can be combined with space-time methods presented in:

• U. Langer, S. E. Moore, M. Neumüller, Space-Time Isogeometric Analysis of Parabolic Evolution
• Equations. Comput. Methods Appl. Mech. Engrg., 306:342–363, 2016.
• O. Steinbach, Space-Time Finite Element Methods for Parabolic Problems, 15(4): 2015, pp. 551–566.

...

Further work: Error estimates for 1 general moving spatial domains, 2 the schemes allowing discontinuity in time, 3 for the domains with complex geometries presented as a

collection of non-overlapping multipatches.

Localized discrete space-time IgA scheme, where h-dependent parameter is replaced by hK-dependent, where K is the element of triangulation Th.

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

29/34 Johann Radon Institute for Computational and Applied Mathematics

Outlook on related technical aspects

1 Establish efficient algorithm of the flux (dual-variable) reconstruction using mixed methods, coarsening meshes and smoothing B-Splines:

• S. K. Kleiss and S. K. Tomar, Guaranteed and sharp a posteriori error estimates in isogeometric

analysis, Computers & Mathematics with Applications 70 (3), 167-190, 2015.

2 Developing fast solver for the systems of structure (Divx Divx + M)Yh = RHS (for the majorant minimization): fast multigrid method and preconditioner:

• J. Kraus and S. Tomar, Algebraic multilevel iteration method for lowest-order Raviart-Thomas

space and applications. Int. J. Numer. Meth. Engng. 86, pp. 1175-1196.

• J. Kraus, R. Lazarov, M. Lymbery, S. Margenov, and L. Zikatanov, Preconditioning

Heterogeneous H(div) Problems by Additive Schur Complement Approximation and Applications, SIAM J. Sci. Comput., 38(2), A875–A898. Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

30/34 Johann Radon Institute for Computational and Applied Mathematics

THANK YOU FOR YOUR ATTENTION!

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

31/34 Johann Radon Institute for Computational and Applied Mathematics

First results: "baby" problem

Ω = (0, 1)2 ⊂ R2, homogeneous Dirichlet BC on ΣD, f = 2 (x1 (1 − x1) + (x2 (1 − x2)) in Ω, u = x1 (1 − x1) x2 (1 − x2), v ∈ S1,1 and y ∈ S1,1 ⊕ S1,1 in Ω. DOFs [e] 2 M2 y − ∇xv2 divxy + f 2 Ieff 25 0.001400290 0.00162422 0.00138683 0.000185002 1.1599 81 0.000347933 0.000376191 0.000347105 1.15488e-05 1.0812 289 8.68498e-05 9.03414e-05 8.68228e-05 6.89776e-07 1.0402 1089 2.17042e-05 2.21382e-05 2.17033e-05 4.25728e-08 1.0200 4225 5.42552e-06 5.54477e-06 5.42421e-06 1.30786e-08 1.0220 16641 1.35635e-06 1.36310e-06 1.35635e-06 1.65615e-10 1.0050 66049 3.39085e-07 3.40956e-07 3.39080e-07 5.11061e-11 1.0055 263169 8.47711e-08 8.50052e-08 8.47708e-08 3.19383e-12 1.0028 1050620 2.11928e-08 2.12613e-08 2.11927e-08 1.09107e-12 1.0032

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

32/34 Johann Radon Institute for Computational and Applied Mathematics

Localized space-time IgA scheme

Assume that p ≥ 2 and m ≤ p − 1 ⇒ V0h ⊂ C1(Q) : V0h ⊂ V 2

0 := H1 0,0(Q) ∩ H2(Q).

Consider each element Ki ∈ Kh, i = 1, ..., NKi ∀u ∈ V 2

0 , ∀vh ∈ V0h:

• ∂tu − ∆xu, vh + δKi ∂tvh
• Ki = (f , vh + δKi ∂tvh)Ki ,

where δKi = θi hKi , θi > 0 and hKi := diam(Ki). Ki Kj Eij = Ki ∩ Kj nij nji Summing up all the elements in Kh:

• Ki ∈Kh
• ∂tu−∆xu, vh)Ki +
• Ki ∈Kh

δKi

• ∂tu−∆xu, ∂tvh
• Ki =
• Ki ∈Kh

(f , vh)Ki +

• Ki ∈Kh

δKi (f , ∂tvh)Ki , ⇓ ⇓ (∂tu−∆u, vh)Q +

• Ki ∈Kh

δKi

• ∂tu−∆xu, ∂tvh
• Ki

= (f , vh)Q +

• Ki ∈Kh

δKi (f , ∂tvh)Ki .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

33/34 Johann Radon Institute for Computational and Applied Mathematics

Localized space-time IgA scheme

Integrate by parts for ∀u ∈ V 2

0 (Q) and ∀vh ∈ V0h:

Ki Kj Eij = Ki ∩ Kj nij nji

• Ki ∈Kh

δKi

• − ∆xu, ∂tvh
• Ki

=

• Ki ∈Kh

δKi

• ∇xu, ∇x(∂tvh)
• Ki −
• Eij ∈EI

h

• ∇xu|Ki · nx

Eij , ∂tvh

• Eij
• =
• Ki ∈Kh

δKi ∇xu, ∇x(∂tvh)

• Ki −
• Eij ∈EI

h

• ([∇xu]Eij , ∂tvh)Eij +

(δKi −δKj ) δKi

(∇xu|Ki · nx

Eij , ∂tvh)Eij

• ,

where nx

Eij is a projection of nEij to the plane Rd−1 x

, nEij is the external normal to Eij = Ki ∩ Kj ∈ EKi

h , and

[∇xu]Eij = (∇xu|Ki − ∇xu|Kj ) · nx

Eij .

The term

• [∇xu]Eij , ∂tvh
• Eij \∂Q vanishes due to the extra smoothness of u ∈ V 2

0 (Q). Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations

34/34 Johann Radon Institute for Computational and Applied Mathematics

Localized space-time IgA scheme

Localized discrete scheme: ah,loc(uh, vh) = lh,loc(vh), ∀vh, uh ∈ Vh, (1) where ah,loc(uh, vh) := (∂tu, vh)Q + (∇xu, ∇xvh)Q +

• Ki ∈Kh

δKi

• (∂tu, ∂tvh)Ki + (∇xu, ∇x(∂tvh))Ki
• −
• Ki ∈Kh
• Eij ∈EKi

h ⊂EI h

(δKi − δKj )

• ∇xu|Ki · nx

Eij , ∂tvh

• Eij

and lh,loc(vh) := (f , vh)Q +

• Ki ∈Kh

δKi (f , ∂tvh)Ki .

Model I-BVP problem Majorants for model I-BVP problem Majorant for space-time IgA scheme Conclutions www.ricam.oeaw.ac.at Svetlana Matculevich, Functional a posteriori error estimates for space-time IgA approximations