Error estimates of some non linear problems Vanessa Lleras Institut - - PowerPoint PPT Presentation

Error estimates of some non linear problems

Vanessa Lleras Institut montpelli´ erain Alexander Grothendieck UMR CNRS 5149 CEMRACS 2015 July 29, 2015

1 Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM

◮A priori error estimates ◮A posteriori error estimates

2 Error estimates for nonlinear eigenvalue problem

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

The continuous problem

The problem of homogeneous isotropic linear elasticity : with f ∈ (L2(Ω))2, let u ∈ (H1(Ω))2 the displacement field solution of        −div σ(u) = f in Ω, σ(u) = Aε(u) on Ω, u = 0 on ΓD, σ(u)n = 0 on ΓN.

• u is the displacement field,
• ε(u) = (∇u +t ∇u)/2 is the linearized strain tensor field,
• σ is the stress tensor field,
• A is the Hooke’s tensor,
• f = (f1, f2) ∈ (L2(Ω))2 represents the volume forces,
• n denotes the normal unit outward vector of Ω on ∂Ω.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Setting of the problem

• The problem is to find u ∈ V = {v ∈ (H1(Ω))2; v = 0 on ΓD} such

that

• Ω

σ(u) : ε(v) dx =

• Ω

f .v dx, ∀v ∈ V

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Setting of the problem

• The problem is to find u ∈ V = {v ∈ (H1(Ω))2; v = 0 on ΓD} such

that

• Ω

σ(u) : ε(v) dx =

• Ω

f .v dx, ∀v ∈ V

• We approximate the continuous problem by a finite element method defined
• n a regular family of triangulations of the domain.

Let Th be the partition of ¯ Ω into elements. The family of triangulations Th, h > 0 of Ω which satisfies the following conditions :

• any 2 triangles in Th share at most a common edge or a common

vertex (in 2D),

• the minimal angle of all triangles in the whole family Th is bounded

away from zero. is called regular

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Setting of the problem

• The problem is to find u ∈ V = {v ∈ (H1(Ω))2; v = 0 on ΓD} such

that

• Ω

σ(u) : ε(v) dx =

• Ω

f .v dx, ∀v ∈ V

• We approximate the continuous problem by a finite element method defined
• n a regular family of triangulations of the domain.

Let Th be the partition of ¯ Ω into elements. The family of triangulations Th, h > 0 of Ω which satisfies the following conditions :

• any 2 triangles in Th share at most a common edge or a common

vertex (in 2D),

• the minimal angle of all triangles in the whole family Th is bounded

away from zero. is called regular

• The discrete problem is to find uh ∈ Vh the unique solution of
• Ω

σ(uh) : ε(vh) dx =

• Ω

f .vh dx, ∀vh ∈ Vh.

• Existence and uniqueness of the problem thanks to Lax Milgram

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Error estimates

Given a norm ., an approximation η to an error e = u − uh is called an error estimator. We can distinguish 2 type of errors :

• a priori estimates : allow to qualify the tendency of the

approximation properties as a function of the number of degrees

• f freedom and the amount of work necessary for the

computation of the discrete solution.

• a posteriori estimates : provide a precise upper bound of the

actual error after a computation has been performed. The a posteriori indicators may tell you what to do to improve the accuracy.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

A priori error estimators

We bound the error by a constant (not fully known) times the best approximation given by the projection of the exact solution onto the discrete space : If u lies in Hs+1(Ω), 0 ≤ s ≤ l with Pl finite elements then u − uhH1(Ω) ≤ chsuHs+1(Ω) which means that the method is convergent of order s.

• give asymptotic rates of convergence as the mesh parameter h

tends to zero

• give information about stability of various solvers
• require regularity conditions of the solution which are in general

not available (because of singularities)

• based on the stability properties of the discrete operator
• insufficient since they only yield information on the asymptotic

behavior.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

A posteriori estimates

An a posteriori error estimation verifies : ◮ A global upper bound : u − uh2

1,Ω ≤ C

• T∈Th

ηT(uh)2 ◮ A local lower bound ηT(uh)2 ≤ CT

• T’ near T

u − uh2

T ′

◮ Asymptotic exactness :

• T∈Th ηT(uh)2

u − uh2

1,Ω

tends to 1 when the mesh size converges to zero

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

A posteriori error estimators

The idea of a posteriori estimation is to determine the order of the error without knowing the exact solution of the problem.

• can be extracted from the numerical solution and the given data
• f the problem which make them computable
• are less expensive to calculate than the computation of the

numerical solution

• are based on the stability properties of the continuous operator
• have global upper bounds which are sufficient to obtain a

numerical solution with the accuracy below a prescribed tolerance.

• employ information about the continuous problem.
• can evaluate the quality of the finite element computations by

locating the zones where the error is important and we can couple these informations with a mesh adaptivity technique which provides the user with the desired quality and minimizes the computation costs

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Residual techniques

The residual technique was introduced by Babuska and Rheinboldt in 1978. The residual is defined by : (Rh, v) = a(uh, v) − (f , v) ∀v ∈ V We have (Rh, v) = a(uh − u, v − vh) Therefore the error of the approximation is determined by the residual norm : RhH−1 = supv=0 (Rh, v) vV The error estimate is based on the residual norm estimates.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Residual techniques

We have a(u − uh, v) =

• T∈Th

(

• T

(f + div σ(uh))(v − vh) − 1 2

• E∈E int

T ∪E N T

• E
• σ(uh)n
• E(v − vh))

And a is elliptic. So u − uhH1 ≤ c sup

v∈H1

0(Ω)

a(u − uh, v) vH1 and a(u − uh, v) =≤

• T∈Th

(f + div σ(uh)L2(T)v − vhL2(T) + 1 2

• E∈E int

T ∪E N T

• σ(uh)n
• EL2(E)v − vhL2(E))

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Clement quasi-interpolation operator

We choose vh = Rhv For all v ∈ H1

0(Ω), the operator Rh has the following

approximation properties : v − RhvL2(T) ≤ c1hTvH1(ωT ) and v − RhvL2(E) ≤ c2h1/2

E vH1(ωE )

The constants c1 and c2 are difficult to evaluate and hT is the diameter of a triangle T, hE is the length of an edge E.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Estimation a posteriori

Definition of the residual error estimators for the elasticity problem

Let T ∈ Th. The local residual error estimator is defined by : η1T = hT f + div σ(uh)T , η2T = h1/2

E

  

• E∈Eint

T ∪EN T

• σ(uh)n
• E 2

E

  

1/2

,

Upper error bound

Let u ∈ V be the solution of the continuous problem and let uh ∈ Vh be the solution of the discrete problem. Then u − uhH1  

T∈Th

η2

1T + η2 2T

 

1/2

.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Local error bound

For the local error bound, we use bubble functions and inverse inequalities : For all polynomials v ∈ Pr(T) cvL2(T) ≤ vψ1/2

T L2(T) ≤ c′vL2(T)

and vH1(T) ≤ ch−1

T vL2(T)

For each estimator :

Lower error bound

Let u ∈ V be the solution of the continuous problem and let uh ∈ Vh be the solution of the discrete problem. η1T u − uh1,ωT η2T u − uh1,ωT

1 Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM

◮A priori error estimates ◮A posteriori error estimates

2 Error estimates for nonlinear eigenvalue problem

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

The continuous problem

The problem of homogeneous isotropic linear elasticity : with f ∈ (L2(Ω))2, let u ∈ (H1(Ω))2 the displacement field solution of        −div σ(u) = f in Ω, σ(u) = Aε(u) on Ω, u = 0 on ΓD, σ(u)n = g on ΓN.

• u is the displacement field,
• ε(u) = (∇u +t ∇u)/2 is the

linearized strain tensor field,

• σ is the stress tensor field,
• A is the Hooke’s tensor,
• f = (f1, f2) ∈ (L2(Ω))2 represents the

volume forces,

• g = (g1, g2) ∈ (L2(ΓN))2 represents

the surface loads imposed on ΓN,

• n denotes the normal unit outward

vector of Ω on ∂Ω.

Γ Γ

Ω

Γ

N D F

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

XFEM method

The XFEM was introduced by Mo¨ es, Dolbow and Belytschko in 1999 to bypass the difficulties of modeling fracture problems by FEM ◮ with the classical finite element method : Drawbacks :

• the mesh coincides with the geometry of the crack, difficulties in 3D
• fine mesh at the crack tip,
• high computational cost for the propagation of the crack.
• evolving contat surfaces with the finite element method is

cumbersome due to the need to update the mesh topology to match the geometry of the contact surface

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

XFEM method

The XFEM was introduced by Mo¨ es, Dolbow and Belytschko in 1999 to bypass the difficulties of modeling fracture problems by FEM ◮ with the classical finite element method : Drawbacks :

• the mesh coincides with the geometry of the crack, difficulties in 3D
• fine mesh at the crack tip,
• high computational cost for the propagation of the crack.
• evolving contat surfaces with the finite element method is

cumbersome due to the need to update the mesh topology to match the geometry of the contact surface

◮ with the eXtended Finite Element Method : Advantages :

• mesh independent of the crack path.
• eliminate the need to remesh in some cases
• facilitate adaptivity

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

The XFEM method

An enrichment of the finite element basis localized with shape functions :

• using a Heaviside function for the nodes whose patch is cut by the crack :

H(x) = +1 if (x − x∗). n ≥ 0, and − 1

• therwise,

where x∗ denotes the crack tip and n is a normal to the crack.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

The XFEM method

An enrichment of the finite element basis localized with shape functions :

• using a Heaviside function for the nodes whose patch is cut by the crack :

H(x) = +1 if (x − x∗). n ≥ 0, and − 1

• therwise,

where x∗ denotes the crack tip and n is a normal to the crack.

• using non smooth functions in polar coordinates at the crack tip :

{Fj(x)}1≤j≤4 = { √r sin θ 2 , √r cos θ 2 , √r sin θ 2 sin θ, √r cos θ 2 sin θ }.

−0.5 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 F1 −0.6 −0.4 −0.2 0.2 0.4 0.6 −0.5 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 F2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −0.5 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 F3 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 −0.5 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 F4 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

XFEM method with cut-off function

• 1973 Strang and Fix have introduced and analyzed the asymptotic behavior

near a corner via a cut-off function.

• 2008 Chahine, Laborde and Renard : introduction of a variant of the

XFEM : XFEM enriched by a cut-off function Principle : to enrich a whole area around the crack tip by using a cut-off function.

Definition

χ ∈ C 2(¯ Ω) is a cut-off function such that 0 < r0 < r1 and r denotes the distance to crack tip : χ(r) =    1 if r ≤ r0, χ(r) ∈ (0, 1) if r0 < r < r1, if r ≥ r1.

−0.5 0.5 −0.5 0.5 0.2 0.4 0.6 0.8 1 axe des x axe des y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Advantages :

• reduce the computational cost,
• improve the performances in terms of convergence of the original method.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Contact problems with XFEM

With the XFEM method, two mainly methods have been used to formulate contact problems :

• the penalty approach : Owen, Peric (1982), Khoei (2006),

Dolbow. Drawbacks :

• consideration of an approximation of the contact problem : don’t give

an exact solution,

• the normal contact force is related to penetration by a penalty

parameter.

• Lagrange multiplier method : Gallego (1989), G´

eniaut (2007), Pierres (2010). Advantages :

• stability is improved without compromising the consistency of the

method

• multipliers restricted to the crack

Drawback : vulnerability of the algorithm to solve problem with slipping

contact,

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Contact problems with XFEM

With the XFEM method, two mainly methods have been used to formulate contact problems :

• the penalty approach : Owen, Peric (1982), Khoei (2006),

Dolbow. Drawbacks :

• consideration of an approximation of the contact problem : don’t give

an exact solution,

• the normal contact force is related to penetration by a penalty

parameter.

• Lagrange multiplier method : Gallego (1989), G´

eniaut (2007), Pierres (2010). Advantages :

• stability is improved without compromising the consistency of the

method

• multipliers restricted to the crack

Drawback : vulnerability of the algorithm to solve problem with slipping

contact,

• All these works require an inf-sup condition of

Babuˇ ska-Brezzi for the good approximation of the solution.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

The continuous problem

The problem of homogeneous isotropic linear elasticity : with f ∈ (L2(Ω))2, let u ∈ (H1(Ω))2 the displacement field solution of        −div σ(u) = f in Ω, σ(u) = Aε(u) on Ω, u = 0 on ΓD, σ(u)n = g on ΓN.

• u is the displacement field,
• ε(u) = (∇u +t ∇u)/2 is the

linearized strain tensor field,

• σ is the stress tensor field,
• A is the Hooke’s tensor,
• f = (f1, f2) ∈ (L2(Ω))2 represents the

volume forces,

• g = (g1, g2) ∈ (L2(ΓN))2 represents

the surface loads imposed on ΓN,

• n denotes the normal unit outward

vector of Ω on ∂Ω.

Γ Γ

Ω

Γ

N D F

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

The elastostatic contact problem

Γ

Ω

Γ

D N

Γ

C−

Γ

C+

Figure: The cracked domain Ω

The conditions describing the unilateral contact are formulated by : [un] ≤ 0, σ+

n (u) = σ− n (u) = σn(u)

≤ 0, σn(u) · [un] = 0. The absence of friction tangential forces is given by : σ+

t (u) = σ− t (u) = 0.

We have defined n+ = −n− the normal unit outward vector on ΓC+, [un] is the jump of the normal displacement across the crack ΓC and σ+

t (u) = σ(u)n+ − σ+ n (u)n+ and σ− t (u) = σ(u)n− − σ− n (u)n−.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Variational formulation

The mixed variational formulation of the continuous problem consists in finding (u, λ) ∈ V × M− such that :

       a(u, v) −

• ΓC

λ[vn] dΓ =

• Ω

f.v dΩ +

• ΓN

g.v dΓ, ∀v ∈ V,

• ΓC

(ν − λ)[un] dΓ ≥ 0, ∀ν ∈ M−,

where V =

• v ∈ (H1(Ω))2; v = 0 on ΓD
• and

a(u, v) =

• Ω

σ(u) : ε(v) dΩ. We introduce

M− =

• ν ∈ H− 1

2 (ΓC ) :

• ν, ψ
• − 1

2 , 1 2 ,ΓC

≥ 0 for all ψ ∈ H

1 2 (ΓC ), ψ ≤ 0 a.e. on ΓC

• .
• Existence and uniqueness of the solution proved by Haslinger,

Hlav´ aˇ cek and Neˇ cas. The displacement field can be written : u = ur + us where ur ∈ H2+η(Ω) for a fixed η > 0 and us ∈ H3/2−ǫ(Ω) for a fixed ǫ > 0.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Stabilized method

◮ A lot of finite element methods (with Lagrange multipliers, augmented Lagrangian) requires the verification of a uniform inf-sup condition : ∃β > 0, inf λh∈Mh(µλhn) sup

vh∈Vh

b(λh, vh) λh−1/2,ΓC vh1,Ω ≥ β where Vh and Mh(µλhn) are the finite element space for the displacement and the one for the multiplier and β is a constant strictly positive and independent of the mesh size. ◮ We consider a stabilization method introduced by Barbosa and

• Hughes. This is a mixed finite element method which does not

require an inf-sup condition. The stability of the multiplier is assured by adding supplementary terms in the weak formulation.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Stabilized discrete approximation

• Notations :
• hT : diameter of T and h = maxT∈T h hT.
• ψx : shape function P1 at the node x.
• Nh set of nodes of the triangulation T h.
• N H

h ⊂ Nh set of enriched nodes.

We approximate the continuous problem by a finite element method defined on a regular family of triangulations (T h)h of the uncracked

• domain. We define

Vh =

• vh ∈ (C(Ω))2 : v h

=

• x∈Nh

axψx +

• x∈N H

h

bxHψx + χ

4

• i=1

ciFi

• .

Let x0, ..., xN distinct points lying in ΓC. These nodes form a one dimensional family of meshes of ΓC denoted by T H and H = max

0≤i≤N−1 |xi+1 − xi|. Then we define different choices of nonempty

closed convex set MH−.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Stabilized discrete approximation

The discrete problem is to find uh ∈ Vh and λH ∈ MH− such that

         a(uh, vh) − b(λH, vh) +

• ΓC

γ(λH − Rh(uh))Rh(vh)dΓ = L(vh), ∀ vh ∈ Vh, b(νH − λH, uh) +

• ΓC

γ(νH − λH)(λH − Rh(uh))dΓ ≥ 0, ∀ νH ∈ MH−

where γ is defined to be constant on each element T as γ = γ0hT where γ0 > 0 is a given constant independent of h. Rh is an operator from Vh

• nto L2(ΓC) which approaches the normal component of the stress vector
• n ΓC defined for all T ∈ T h with T ∩ ΓC = ∅ as

Rh(vh)| T∩ΓC =          σn(vh

1),

if | T ∩ Ω1 | ≥ | T | 2 , σn(vh

2),

if | T ∩ Ω2 | > | T | 2 , where vh

1 = vh

| Ω1 and vh

2 = vh

| Ω2 .

• existence and uniqueness of the solution when γ0 is small enough.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

A priori estimates without contact

Chahine, Nicaise, Renard

Assume that the displacement solution u satisfies the smoothness condition u − us ∈ H2(Ω), then u − uhH1 ≤ Chu − χusH2 for P1 finite elements

• definition of an adapted quasi interpolation operator
• definition of different type of triangles : triangles non enriched,

triangles partially enriched by the Heaviside function and triangles totally enriched With these tools, they obtained optimal convergence

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

A priori error estimates

• First contact condition :

W H

0 =

• µH ∈ L2(ΓC) : µH

| (xi ,xi+1) ∈ P0(xi, xi+1), ∀ 0 ≤ i ≤ N − 1

• ,

MH− =

• µH ∈ W H

0 : µH ≤ 0 on ΓC

• .

Theorem

Let (u, λ) be the solution to the continuous problem. Assume that ur ∈ (H2(Ω))2. Let γ0 be small enough and let (uh, λH) be the solution to the discrete problem where MH− = MH− . Then, we have for any η > 0

• u − uh, λ − λH
• hu − χus2,Ω + h1/2H1/2λ1/2,ΓC

+H3/4−η/2(u3/2−η,Ω + λ1/2,ΓC ).

We define for any v ∈ (H1(Ω))2 and any µ ∈ L2(ΓC) the following norms : v = a(v, v)1/2, and |(v, µ)| =

• v2 + γ1/2µ2

0,ΓC

1/2 .

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

A priori error estimates

• Second contact condition :

W H

1 =

• µH ∈ C(ΓC) : µH

| (xi ,xi+1) ∈ P1(xi, xi+1), ∀ 0 ≤ i ≤ N − 1

• .

MH−

1

=

• µH ∈ W H

1 : µH ≤ 0 on ΓC

• .

Theorem

Let (u, λ) be the solution to the continuous problem. Assume that ur ∈ (H2(Ω))2. Let γ0 be small enough and let (uh, λH) be the solution to the discrete problem where MH− = MH−

1

. Then, we have for any η > 0

• u − uh, λ − λH
• hu − χus2,Ω + (H

1−η 2

+ h1/2)λ1/2,ΓC +H

1−η 2 u3/2−η,Ω.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

A priori error estimates

• Third contact condition :

W H

1 =

• µH ∈ C(ΓC) : µH

| (xi ,xi+1) ∈ P1(xi, xi+1), ∀ 0 ≤ i ≤ N − 1

• .

MH−

1,∗ =

• µH ∈ W H

1 :

• ΓC

µHψHdΓ ≥ 0, ∀ ψH ∈ MH−

1

• .

Theorem

Let (u, λ) be the solution to the continuous problem. Assume that ur ∈ (H2(Ω))2. Let γ0 be small enough and let (uh, λH) be the solution to the discrete problem where MH− = MH−

1,∗ .

Then, we have for any η > 0

• u − uh, λ − λH
• hu − χus2,Ω + (h1/2 + H3/2−η)λ1/2,ΓC

+h−1/2H1−ηu3/2−η,Ω.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

A matrix formulation of the Barbosa-Hughes contact problem

The algebraic formulation of the discrete problem is given by :      Find U ∈ RN and L ∈ M

H− such that

(K − Kγ)U − (B − Cγ)TL = F, (L − L)T((B − Cγ)U + DγL) ≥ 0, ∀L ∈ M

H−,

where K, B, Kγ, Cγ, Dγ are the matrices corresponding to the terms a(uh, vh), b(λH, vh),

• ΓC γRh(uh)Rh(vh) dΓ,
• ΓC γλHRh(vh) dΓ,
• ΓC γλHµH dΓ, respectively.

The last inequality can be expressed as an equivalent projection L = P

MH−(L − r((B − Cγ)U + DγL)),

where r is a positive augmentation parameter.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

A matrix formulation of the Barbosa-Hughes contact problem

This last step transforms the contact condition into a nonlinear equation and we have to solve the following system :        Find U ∈ RN and L ∈ M

H− such that

(K − Kγ)U − (B − Cγ)TL − F = 0, −1 r

• L − P

MH−(L − r((B − Cγ)U + DγL))

• = 0.

This allows us to use the semi-smooth Newton method to solve this problem.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Numerical results

The assumptions are :

• domain :¯

Ω = [0, 1] × [−0.5, 0.5]

• crack : ΓC = ]0, 0.5[ × {0}
• we impose a body vector force
• we impose Neumann boundary

conditions

• 3 degrees of freedom are blocked

to eliminate rigid motions

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Numerical results

Figure: Normal contact stress Figure: Von Mises stress

The normal contact stress and the Von Mises stress are not singular at the crack lips.

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Convergence analysis of the non stabilized case

Figure: Error in L2(Ω)-norm of the displacement Figure: Error in H1(Ω)-norm of the displacement Figure: Error in L2(ΓC)-norm of the contact stress

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Convergence analysis of the stabilized case

Figure: Error in L2(Ω)-norm of the displacement Figure: Error in H1(Ω)-norm of the displacement Figure: Error in L2(ΓC)-norm of the contact stress

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Influence of the stabilization parameter

Figure: in L2(ΓC)-norm of the contact stress with P1/P0 elements Figure: in L2(ΓC)-norm of the contact stress with P2/P1 elements Figure: Influence of the stabilization parameter for P1/P0 method

Error estimates for contact problem

◮Linear elasticity without contact and without XFEM ◮Frictionless contact problem with XFEM ◮A priori error estimates ◮A posteriori error estimates

Error estimates for nonlinear eigenvalue problem

Estimation a posteriori without contact

Definition of the residual error estimators for the elasticity problem

Let G ∈ Gh and T ∈ Th be the triangle contening G. The local residual error estimator is defined by : η1G = hT C(hT )f + div σ(χuh,s)G , η2G = h1/2

T

D(hT )   

• E∈Eint

G ∪EN G ∪EF G

• σ(uh)n
• E 2

E

  

1/2

, where C(hT ) =

• − ln(hT ) for the elements in case (iv) of Lemma 1, else

C(hT ) = 1 and D(hT ) =

• − ln(hT ) for the elements in case (iv) of Lemma 2 or

in case (ii) of Lemma 3, otherwise D(hT ) = 1.

Upper error bound

Let u ∈ V be the solution of the continuous problem and let uh ∈ Vh be the solution of the discrete problem. Then u − uhH1  

G∈Gh

η2

1G + η2 2G

 

1/2

.

1 Error estimates for contact problem 2 Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case

A priori estimates A posteriori estimates

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Eigenvalue problem

Let Ω be a bounded Lipschitz domain in Rd and A is a (non-) selfadjoint second order elliptic operator. The classical formulation of the eigenvalue problem is : find u ∈ V = H1

0(Ω) and λ ∈ R with uL2(Ω) = 1 such that

Au = λu in Ω Then the eigenpair (λ, u) ∈ R × V satisfies the variational formulation a(u, v) = λ

• Ω

uv ∀v ∈ V with uL2(Ω) = 1 where a : V × V → F is a bilinear form symmetric generated by A and is assumed to be bounded in V and V-elliptic : |a(u, v)| ≤ C1uV vV ∀u, v ∈ V , a(u, u) ≥ Cu2

V

∀u ∈ V with C1, C > 0 and where .V ⋍ a(., .)1/2 We have the existence and uniqueness of (λ, u) with Lax-Milgram lemma

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Discrete eigenvalue problem

The numerical analysis of these linear eigenvalue problems has been studied particularly with the finite element method (Babuska and Osborn, Strang and Fix...) Let Pp denote the set of continuous piecewise polynomial functions of total degree p ≥ 1, which vanish on the boundary

• f Ω.

For the finite dimensional subspace V p

h (Ω) = {v ∈ V , v|T ∈ Pp, ∀T ∈ Th} ⊂ V , we use the

Ritz-Galerkin discretization : Determine a non-trivial eigenpair (λh, uh) ∈ R × V p

h with

uhL2(Ω) = 1 such that a(uh, vh) = λh

• Ω

uhvh ∀vh ∈ V p

h

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

The algebraic eigenvalue problem

It is known that the continuous problem has a countable set of real eigenvalues 0 < λ1 ≤ λ2 ≤ ... diverging to ∞ and corresponding eigenfunctions u1, u2... bounded in H2(Ω) The discrete eigenvalue problem has a finite set (n=dim V p

h ) of

eigenvalues 0 < λ1,h ≤ λ2,h ≤ ... ≤ λn,h and corresponding eigenvectors u1,h, u2,h..., un,h Let {φh

1, ..., φh n} be a basis for a finite dimensional space V p h . So

uh =

n

• i=1

uh,iφh

i

It follows from the Courant-Fischer min-max theorem that λi ≤ λi,h for all i=1..n

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

A priori error estimates

For all u ∈ H0

1(Ω) ∩ Hl(Ω) , 1 ≤ l ≤ k + 1 with Pk finite

element discretization |λh,1 − λ| ≤ Cuh,1 − u2

H1

There exist h0 and C ∈ R+ such that for all 0 ≤ h ≤ h0 uh,1 − uH1 ≤ Chl−1uHl, uh,1 − uL2 ≤ Chuh,1 − uH1

1 Error estimates for contact problem 2 Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case

A priori estimates A posteriori estimates

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Examples of non linear eigenvalue problems

Non-linear eigenvalue problems are involved in many application fields such as :

• Mechanics : vibration modes within nonlinear elasticity
• Physics : steady states of Bose Einstein condensates :

Gross-Pitaevskii equation.

• Chemistry and electronic structure calculations :

Hartree-Fock model.

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Setting of the problem

We consider the minimization problem I = inf {E(v), v ∈ X,

• Ω

v2 = 1} with E(v) = 1 2

• Ω

|∇v|2 + 1 2

• Ω

V |v|2 + µ 2

• Ω

|v|4 where Ω = (0, L)d and the Sobolev space X = {v|Ω, v ∈ H1

loc(Rd) such that v is periodic } or H1 0(Ω)

and µ ≥ 0. V is the potential function ∈ Lp(Ω) for some p > max(1, d/2).

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Setting of the problem

The problem can be seen as the Euler-Lagrange equations associated with the constraint minimization problem. Then the eigenfunctions are the critical points of the energy functional

• ver the unit sphere.

For a non rotating problem, we have that the global minimal solution is unique (Lieb, Seiringer, Yngvason (2000)) and gives a ground state.

• The minimization problem has exactly 2 minimizers u and
• u
• u is the ground state of the non linear eigenvalue problem

−∆u + Vu + µu3 = λu, uL2 = 1

• u ∈ C 0,α(¯

Ω) for some α > 0 and u > 0 in Ω

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Variational approximation

The variational approximation of the minimization problem in XN consists in solving IN = inf {E(vN), vN ∈ XN,

• Ω

v 2

N = 1}

• There exists at least one minimizer uN such that
• Ω uuN ≥ 0

which satisfies ∀vN ∈ XN,

• Ω

∇uN∇vN +

• Ω

VuNvN + µ

• Ω

u3

NvN = λN

• Ω

uNvN for some λN ∈ R.

• The minimizer uN is unique for N small enough and there exists

0 < C < ∞ such that u − uNH1 ≤ CminvN∈XNvN − uH1

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

A priori error estimates

Linear case : µ = 0 (Babuska and Osborn 1991)

There exists 0 < c ≤ C < ∞ such that for all N > 0 u − uNH1 ≤ CminvN∈XNvN − uH1 cu − uN2

H1 ≤ E(uN) − E(u) ≤ Cu − uN2 H1

|λ − λN| ≤ Cu − uN2

H1

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

A priori error estimates

The numerical analysis of such non-linear eigenvalue problems is quite recent :

Zhou 2004

There exists 0 < N0 and C ∈ R+ such that for all 0 < N < N0 u − uNH1 ≤ CminvN∈XN vN − uH1 |λ − λN| ≤ Cu − uN2

H1 + µ

• Ω

|u2

N(uN + u)(uN − u)|

≤ Cu − uN2

H1 + u − uNL2

Canc` es, Chakir, Maday, 2010

cu − uN2

H1 ≤ E(uN) − E(u) ≤ Cu − uN2 H1

u − uN2

H1 ≤ Cu − uNH1minψN∈XN ψuN−u − ψNH1

where ψuN−u ∈ u⊥ = {v ∈ H1

0(Ω) such that (v, u) = 0} is the unique

solution to the adjoint problem ∀v ∈ u⊥, < (E ′′(u) − λ)ψuN−u, v >X ′,X=< uN − u, v >X ′,X

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Planewave discretisation

Planewave basis sets : for all v ∈ L2(Ω), we have v(x) =

• k∈Z d
• vkek(x), where ek(x) =

eikx (2π)d/2 , k ∈ Zd with vk = (2π)−d/2

Ω v(x)e−ikxdx

We choose XN = Span{ek, |k|∞ ≤ N} with vHs = (

• k∈Zd,|k|≤N

(1 + |k|2)s| vk|2)1/2 For all v ∈ Hs

per(Ω), the best approximation of v in Hs per(Ω) for

any r ≤ s is ΠNv =

• k∈Zd,|k|∞≤N
• vkek(x)

and ∀v ∈ Hs

per(Ω), v − ΠNvHr ≤

1 Ns−r vHs

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Planewave discretization

A priori error estimates (Chakir, Canc` es, Maday)

Assume that u ∈ Hs

per(Ω), for some s > d/2 then (uN)

converges to u in Hs

per(Ω) and there exists 0 < c ≤ C < ∞

such that for all n ∈ N uN − uHs ≤ C Ns−b uHs, ∀ − s + 2 ≤ b < s |λ − λN| ≤ C N2s uHs equivalent to the linear case

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Finite element discretization

Let (Th) be a family of regular triangulations of Ω and such that

• Ω uuh,k ≥ 0

With P1 finite element discretization

|λh,1 − λ| ≤ C(uh,1 − u2

H1 + uh,1 − uL2)

There exist h0 and C ∈ R+ such that for all 0 ≤ h ≤ h0 uh,1 − uH1 ≤ Ch, uh,1 − uL2 ≤ Ch2, |λh,1 − λ| ≤ Ch2

With P2 finite element discretization

|λh,2 − λ| ≤ C(uh,2 − u2

H1 + uh,2 − uH−1)

There exist h0 and C ∈ R+ such that for all 0 ≤ h ≤ h0 uh,2 − uH1 ≤ Ch2, uh,2 − uL2 ≤ Ch3, |λh,2 − λ| ≤ Ch4

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Finite element discretization

Numerical test with Ω = (0, π)2 and V (x1, x2) = x2

1 + x2 2

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Setting of the problem

We provide an a posteriori error analysis for variational approximations of the ground state eigenvector of a non linear elliptic problem of the Gross-Pitaevskii type : −∆u + Vu + u3 − αLzu = λu uL2 = 1 with periodic boundary conditions in dimension d=2 or 3. α is the angular velocity of the condensate along the z-axis and the operator Lz is such that Lz = −i(xδy − yδx) The wave function u is normalized which corresponds to the mass conservation constraint. Bao and Cai has shown that ground states for rotating gases can be obtained for α small enough. It’s not the case when the rotation speed is too large.

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Setting of the problem

We focus on the following non-linear eigenvalue problems arising in the study of variational problems of the form I = inf {E(v), v ∈ X,

• Ω

v2 = 1} where Ω = (0, 2π)d and the Sobolev space X = {v|Ω, v ∈ H1

loc(Rd) such that v is 2π − periodic }

provided with the norm .H1. The energy functional E is of the form : E(v) = 1 2

• Ω

|∇v|2 + 1 2

• Ω

V |v|2 + 1 4

• Ω

|v|4 −

• Ω

α¯ vLzv

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Planewave discretisation

Planewave basis sets : for all v ∈ L2(Ω), we have v(x) =

• k∈Z d
• vkek(x), where ek(x) =

eikx (2π)d/2 , k ∈ Zd with vk = (2π)−d/2

Ω v(x)e−ikxdx

We choose XN = Span{ek, |k|∞ ≤ N} with vHs = (

• k∈Zd,|k|≤N

(1 + |k|2)s| vk|2)1/2 (uN, λN) is the variational approximation of the ground state eigenpair (u, λ) based on a Fourier spectral approximation.

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Iterative process

(uk

N, λk N) is the approximate solution at the kth iteration of the

algorithm used to solve the non-linear problem. The algorithm used to solve the equation numerically in the space XN is the following :

• starting from a given couple (u0

N, λ0 N)

• we solve at each step the linear equation

ΠN(−∆uk⋆

N + Vuk⋆ N + |uk−1 N

|2uk⋆

N − αLzuk⋆ N ) = λk−1 N

uk−1

N

We find uk⋆

N which is a non-normalized vector.

• Therefore we write uk

N =

uk⋆

N

uk⋆

N L2

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Iterative process

Finally we define the eigenvalue as a Rayleigh quotient being λk

N =

• Ω |∇uk⋆

N |2 +

• Ω V |uk⋆

N |2 +

• Ω |uk⋆

N |4 −

• Ω α ¯

uk⋆

N Lzuk⋆ N

• Ω |uk⋆

N |2

which is also λk

N =

• Ω |∇uk

N|2 +

• Ω V |uk

N|2 +

• Ω |uk

N|4 −

• Ω α ¯

uk

NLzuk N

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

A posteriori estimates

For the global error Rk

N = −∆uk N + Vuk N + (uk N)3 − αLzuk N − λk Nuk N,

we divide it into 2 residues characterizing the error due to the discretization of the space and the finite number of iterations when solving the problem numerically. We obtain the error due to the discretization dimension (based on the numerical scheme) RdiscH−1 = −∆uk

N+Vuk N+|uk−1 N

|2uk

N−αLzuk N−uk⋆ N −1 L2 λk−1 N

uk−1

N

H−1 and the error due to the number of iterations : RitH−1 = (uk

N)3 − |uk−1 N

|2uk

N − λk Nuk N + uk⋆ N −1 L2 λk−1 N

uk−1

N

H−1

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

A posteriori estimates

We provide a posteriori indicators based on residual techniques that discriminate the effect of the discretization parameter (number of degrees of freedom) from the parameters attached to the solution procedure (number of iterations).

upper bound

With 0 < θ ≤ 1 and C > 0, we obtain the following upper bound : u − uk

NH1 = maxv∈H1

per

• Ω ∇(u − uk

N)∇v +

• Ω(u − uk

N)v

uH1 ≤ θ(RdiscH−1 + CRitH−1)

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Numerical simulations

The numerical simulations are performed with Freefem++ in dimension 1 without rotation and with a potential V given by its Fourier coefficients ˆ Vk = 1 √ 2π 1 |k|4 − 1/4 So V ∈ Lp for any p > 1

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

With large number of iterations

The error components evolution with k=32 :

N err disc err iter err total u − ueH1 15 1.302454e-06 9.907161e-13 0.0002404655 1.295024e-06 20 3.533197e-07 9.91412e-13 6.523161e-05 3.522857e-07 25 1.288206e-07 9.626073e-13 2.378349e-05 1.286171e-07 30 5.656094e-08 9.853153e-13 1.044256e-05 5.651427e-08 35 2.821977e-08 9.106983e-13 5.210072e-06 2.82096e-08 40 1.545719e-08 9.275063e-13 2.853782e-06 1.545635e-08 45 9.091335e-09 9.200662e-13 1.678487e-06 9.092774e-09 50 5.65577e-09 1.01678e-12 1.044196e-06 5.657532e-09 55 3.681764e-09 9.744701e-13 6.797458e-07 3.683332e-09 60 2.488163e-09 1.281291e-12 4.59377e-07 2.48944e-09 65 1.735199e-09 1.001215e-12 3.203609e-07 1.736207e-09 70 1.242906e-09 9.20199e-13 2.294715e-07 1.243696e-09 75 9.110445e-10 1.02119e-12 1.682017e-07 9.116627e-10 80 6.813302e-10 9.679636e-13 1.257905e-07 6.818161e-10 85 5.186014e-10 1.576103e-12 9.574673e-08 5.189856e-10 90 4.009506e-10 1.011194e-12 7.402554e-08 4.012562e-10 95 3.143366e-10 9.908568e-13 5.803436e-08 3.145809e-10 100 2.495311e-10 9.29715e-13 4.606966e-08 2.497272e-10

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

In large dimensional space

The error components evolution with N=100 :

k err disc err iter err total u − ueH1 1 0.06049483 11.78493 21.1699 0.08742198 3 0.003971473 1.863977 2.590739 0.01101804 5 0.0005223785 0.2637348 0.3600835 0.001545706 7 7.332219e-05 0.03738365 0.05091903 0.000219117 9 1.038476e-05 0.005302901 0.007220152 3.109144e-05 11 1.472962e-06 0.0007523643 0.001024309 4.411748e-06 13 2.089799e-07 0.0001067494 0.0001453323 6.259884e-07 15 2.965225e-08 1.514639e-05 2.062079e-05 8.882132e-08 17 4.214527e-09 2.149089e-06 2.926193e-06 1.260514e-08 19 6.469966e-10 3.049304e-07 4.176891e-07 1.805515e-09 21 2.635141e-10 4.3266e-08 7.477994e-08 3.559949e-10 23 2.498203e-10 6.138883e-09 4.682161e-08 2.523072e-10 25 2.495369e-10 8.709678e-10 4.608491e-08 2.497792e-10 27 2.495312e-10 1.236383e-10 4.607004e-08 2.497282e-10 29 2.495311e-10 1.752203e-11 4.606965e-08 2.497272e-10 31 2.495311e-10 2.550145e-12 4.606965e-08 2.497272e-10

Error estimates for contact problem Error estimates for nonlinear eigenvalue problem

◮Linear case ◮Nonlinear case A priori estimates A posteriori estimates

Perspectives

• A posteriori error estimates for the contact problem with

XFEM method

• Error estimates with XFEM and Darcy problem.
• Simulations for the non-linear eigenvalue problem with

rotation

• Comparison with a posteriori error estimators using the

finite element discretization and the Sobolev gradient method of Danaila and Kazemi. We define ∇A = ∇ + iαAt with A = (y, −x, 0) and the norm u2

HA = u2 L2 + ∇Au2