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Adaptive FEM, Approach with hp- and Goal-Oriented A Posteriori Error Estimator Arezou Ghesmati Dept. of Mathematics Texas A&M University 5 th deal.II Users and Developers Workshop Aug. 6, 2015 Arezou Ghesmati (TAMU) hp- and Goal-Oriented

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Adaptive FEM, Approach with hp- and Goal-Oriented A Posteriori Error Estimator

Arezou Ghesmati

  • Dept. of Mathematics

Texas A&M University 5th deal.II Users and Developers Workshop

  • Aug. 6, 2015

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Outline

1 Motivation 2 hp-Adaptive Finite Element Method(hp-AFEM)

Adaptivity A Posteriori Error Estimator Reliability and Efficiency of Estimator hp-Adaptive Refinement Loop Contraction Convergence Results

3 Numerical Results 4 Ongoing/Future Research Direction

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Motivation

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Motivation

The Boussinesq Equations: −ν∆u + ∇̺ = f(T, g, C) ∇ · u = 0 ∂u ∂t + u · ∇T − ∇ · κ∇T = γ

https://images.google.com https://aspect.dealii.org Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM

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Stokes or Creeping Flow

Given f ∈ L2(Ω)d, {d = 2, 3} , ν ≥ 1, consider the Stokes equations as

  • ur model problem: Find u : Ω → Rd and ̺ : Ω → R such that

−ν∆u + ∇̺ = f in Ω ∇ · u = 0 in Ω u = 0

  • n

Γ.

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Weak Formulation

We denote The standard weak formulation of problem; Seek [u, ̺] ∈ H such that L([u, ̺]; [v, q]) = (f, v)Ω ∀[v, q] ∈ H. Where H := H1

0(Ω)d × L2 0(Ω).

the bilinear form L : H × H → R is defined as: L([u, ̺]; [v, q]) := (ν∇u, ∇v)Ω−(̺, ∇·v)Ω−(∇·u, q)Ω ∀[u, ̺], [v, q] ∈ H.

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Discretization

Due to the continuous inf-sup condition inf

[u,̺]∈H sup [v,q]∈H

L([u, ̺]; [v, q]) (∇uΩ + ̺Ω) (∇vΩ + qΩ) ≥ κ > 0, we define the finite element spaces V p

u (T ) and V p ̺ (T ) by

V p

u (T ) :=

  • u ∈ H1

0(Ω) : u|K ◦ TK ∈ QpK

  • ˆ

K

  • for all K ∈ T
  • and

V p

̺ (T ) :=

  • ̺ ∈ L2

0(Ω) : ̺|K ◦ TK ∈ QpK−1

  • ˆ

K

  • for all K ∈ T
  • ,

the discrete approximation is obtained by finding [uFE, ̺FE] ∈ Vp(T ) such that : Vp(T ) := V p

u (T )d × V p ̺ (T )

L ([uFE, ̺FE] ; [vFE, qFE]) = (f, vFE)Ω ∀ [vFE, qFE] ∈ Vp(T ).

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hp-Adaptive Finite Element Method

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Adaptivity

h- Adaptive FEM p- Adaptive FEM hp- Adaptive FEM

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A Posteriori Error Estimator

The idea behind a posteriori error estimation is to access the error between the exact solution and its finite element approximation, in terms of known quantities only! Reliability: ∇ (u − uFE)Ω + ̺ − ̺FEΩ ≤ Crelη (uFE, ̺FE, f) . Local error estimators: η2(uFE, ̺FE, f) =

  • K∈T

η2

K (uFE, ̺FE, f) ,

Computational Efficiency ηK (uFE, ̺FE, f) ≤ Ceff (∇ (u − uFE)K + ̺ − ̺FEK) ∀K ∈ T

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Residual Based A Posteriori Error Estimator

A posteriori error estimator η shall be the sum of local error indicators ηK: η2 :=

  • K∈T

η2

K

Local Error Estimator: The local a posteriori error estimator ηK can be decomposed into cell contribution and interface contribution: η2

K := η2 K;R + η2 K;B,

where ηK;R denotes the residual-based term and ηK;B indicates the jump-based term. These are defined by η2

K;R := h2 K

p2

K

  • IK

pKf + ν∆uFE − ∇̺FE

  • 2

K + (∇ · uFE)2 K

and η2

K;B :=

  • e∈E(K)

he 2pe

  • ν ∂uFE

∂nK

  • 2

e

.

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Reliability and Efficiency of Estimator

Reliability: Let [uFE, ̺FE] ∈ Vp(T ) be the solution of discrete problem and [u, ̺] ∈ H be solution of weak problem. Further, assume that triangulation T is (γh, γp)-regular. Then, there exists some constant Crel > 0 independent of mesh size vector h and polynomial degree vector p such that

∇ (u − uFE)2

Ω + ̺ − ̺FE2 Ω ≤ Crel

  • K∈T
  • p2

Kη2 K + h2 K

p2

K

  • IK

pKf − f

  • 2

K

  • .

Efficiency: Let [uFE, ̺FE] ∈ Vp(T ) be the solution of discrete problem, and [u, ̺] ∈ H be solution of weak problem. Further, we assume that triangulation T is (γh, γp)-regular. Then, there exists some constant Ceff > 0 independent of mesh size vector h and polynomial degree vector p such that

η2

K ≤ Ceff

  • pK
  • ν2 ∇ (u − uFE)2

ωK + ̺ − ̺FE2 ωK

  • + h2

K

p2

K

  • IK

pKf − f

  • 2

ωK

  • for all K ∈ T .

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hp-Adaptive Refinement Loop

The fully automatic hp-adaptive refinement strategy is based on the standard adaptive loop SOLVE − → ESTIMATE − → MARK − → REFINE. SOLVE and REFINE are almost the same in all adaptive refinement patterns. ESTIMATE and MARK are the two most crucial modules in hp-adaptive method.

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hp- Adaptive Refinement Algorithm

Initialization: Set N = 0, a coarse mesh T0, θ ∈ [0, 1] and also tolerance TOL. SOLVE: Find the solution (uFE, ̺FE) of discrete problem. ESTIMATE: Compute a posteriori error estimation. If η < TOL then STOP the algorithm, else, MARK: select set of cells A to be marked either with h- or p-refinement REFINE: Given (A, (jK)K∈A), we refine the cells contained in A with refinement patterns jK corresponding to each cell. Then set N = N + 1 and go to step SOLVE.

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Module MARK

More information needed in module MARK to choose the best adaptive strategy: Convergence Estimator: kK,j, j ∈ {1, 2, 3} j=1 j=2 j=3 Efficiency ≈ Workload number: ̟K,j = n DoFs j ∈ {1, 2, 3} Choose the best refinement pattern ⇒ find integer jK ∈ {1, 2, 3} such that: kK,jK ̟K,jK = max

j∈{1,2,3}

kK,j ̟K,j ,

  • K∈A

k2

K,jkη2 K ≥ θ2η2

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Challenges to calculate the Convergence Estimator: kK,j

Considering the residual of Stokes problem on the local patch domain ωK, we have:

(∇v, ∇(wN,j

u

))ωK + (q, wN,j

̺

)ωK = L([v, q]; [e, E])ωK, ∀(v, q) ∈ Vp

K,j(TN|ωK).

The pair (wN,j

u

, wN,j

̺

) ∈ Vp

K,j(TN|ωK) is defined to be the Ritz representation of the

residual. kK,j = 1 ηK(uFE, ̺FE)

  • ∇wN,j

u

  • 2

ωK

+

  • wN,j

̺

  • 2

ωK

1

2

regular patch non-uniform patch

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Build the local triangulation to compute ⇒ kKj

non-uniform patch get-cells-at-coarsest- common-level set-FE-Nothing

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Contraction Convergence Results

Contraction for Error in Energy Norm: ∇(u−uN+1

FE )2 Ωd+̺−̺N+1 FE 2 Ω ≤ µ

  • ∇(u − uN

FE)2 Ωd + ̺ − ̺N FE2 Ω

  • Quasi- Convergence :

∇(u − uN+1

FE )2 Ωd+̺ − ̺N+1 FE 2 Ω + ϑη2(TN+1) ≤ µ

  • ∇(u − uN

FE)2 Ωd+

̺ − ̺N

FE2 Ω + ϑη2(TN )

  • for constants ϑ > 0 and µ ∈ (0, 1) independent of mesh size h and

polynomial degree vector p.

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Important Equivalence Result

Total Error: ∇(u − uN

FE)2 Ωd + ̺ − ̺N FE2 Ω + osc2 N

Quasi Error: ∇(u − uN

FE)2 Ωd + ̺ − ̺N FE2 Ω + ϑη2(TN )

Quasi Error ≈ η2 ≈ Total Error

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Numerical Results

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Example-1

Manufactured solution on L-shaped domain: Let Ω ∈ R2 be L-shaped domain, (−1, 1)2\([0, 1] × [−1, 0]) we enforce appropriate inhomogeneous boundary conditions for velocity u on Γ such that the analytical solution u : Ω → R2 and ̺ : Ω → R are given by: u = −ex(y cos(y) + sin(y)) exy sin(y)

  • ,

̺ = 2ex sin(y)−(2(1−e)(cos(1)−1))/3.

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hp- adaptive refinement, cycle = 0

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hp- adaptive refinement, cycle = 2

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hp- adaptive refinement, cycle = 4

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hp- adaptive refinement, cycle = 6

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hp- adaptive refinement, cycle = 9

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hp- adaptive refinement, cycle = 12

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Distribution of a posteriori error estimator in hp- adaptive refinement

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Error- Error Estimator in hp-AFEM

Comparison of actual and estimated energy error vs DoFs. Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM

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hp- and h- Error-Estimator

Comparison of the estimated error with hp- and h- adaptive mesh refinement. Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM

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Example-2

Smooth solution in two dimensions: Let Ω : (−1, 1) × (−1, 1) and let velocity u : Ω → R2 and ̺ : Ω → R be give by u = 2y cos(x2 + y2) −2x cos(x2 + y2)

  • ,

̺ = e−10(x2+y2) − ̺m where pm is such that

  • Ω ̺ = 0.

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hp- adaptive refinement, cycles: 0, 2, 4, 5, 7, 8, 9, 11 Min poly. degree=3, Max poly. degree=8

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Distribution of posteriori error estimator in hp- AFEM

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Error- Error Estimator in hp-AFEM

Comparison of actual and estimated energy error vs DoFs. Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM

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hp- and h- Error-Estimator

Comparison of the estimated error with hp- and h- adaptive mesh refinement. Arezou Ghesmati (TAMU) hp- and Goal-Oriented AFEM

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Example-3

Singular solution in two dimensions

We consider the L-shaped domain Ω := (−1, 1)2 ([0, 1] × [−1, 0]) with reentrant angle ω = 3π/2 at the origin. Let α ≈ 0.544 be an approximation of the smallest root of a nonlinear equation: The exact velocity u and pressure ̺ are given in polar coordinates by u(r, ϕ) = rα

  • cos(ϕ)ψ

′(ϕ) + (1 + α) sin(ϕ)ψ(ϕ)

sin(ϕ)ψ

′(ϕ) − (1 − α) cos(ϕ)ψ(ϕ)

  • and

̺(r, ϕ) = −rα−1 (1 + α)2ψ

′(ϕ) + ψ ′′′(φ)

1 − α where ψ(ϕ) is a smooth function

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Example-3, hp- adaptive refinement, cycles: 0, 2, 5, 10, 11, 16, 19, 20

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Distribution of a posteriori error estimator in hp- adaptive refinement

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Example-3

Figure: Comparison of actual and estimated energy error vs DoFs.

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Goal Oriented A Posteriori Error Estimator in h- and hp-AFEM

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Primal-Dual problem

Primal Problem: −ν∆u + ∇̺ = f in Ω ∇ · u = 0 in Ω = ⇒ L([u, ̺]; [v, q]) = (f, v)Ω u = 0

  • n

Γ. Dual Problem: −ν∆zu + ∇z̺ = j(u, ̺) in Ω ∇ · zu = 0 in Ω = ⇒ L([v, q]; [zu, z̺]) = (v, j)Ω zu = 0

  • n

Γ.

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Goal-Oriented A Posteriori Error Estimator

Goal-Oriented estimator: ζ :=

  • K∈T

ζK Goal-Oriented local error estimator ζK are defined as ζK := ρK · ηK(uFE, ̺FE, f) the local weight ρK is given by ρK := ˜ ηK + ∇vu,FE(ωK,2) + v̺,FE(ωK,2)

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Reliability and Efficiency:

Theorem -1:

Reliable Goal-Orineted A Posteriori Error Estimator

|J(u, ̺) − J(uFE, ̺FE)| ≤ Crel

  • K∈T
  • ηK + hK

pK f − IK

pKfK

  • ·
  • ρK + hK

pK j − IK

pKωK,2

  • Theorem -2:

Efficient Goal-Orineted A Posteriori Error Estimator ζK ≤ Ceff

  • pK(∇(u − uFE)ωK,2 + ̺ − ̺FEωK,2) + hK

p

1 2

K

f − IpKfωK,2

  • p

3 4

K(∇(zu − zu,FE)ωK,2 + z̺ − z̺,FEωK,2) + hK

p

1 4

K

j − IpKjωK,2

  • Arezou Ghesmati

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Go-AFEM Algorithm for hp-Refinement

1 Set N = 0, Tol > 0, θ ∈ (0, 1], and initialize coarse grid T0. 2 Solve the Primal problem 3 Solve the Dual problem 4 For every cell K ∈ TN, solve the local variational problem over

patches

5 Compute A Posteriori Error Estimator given as

ζK := ρK · ηK(uFE, ̺FE, f) ρK := ˜ ηK + ∇vu,FE(ωK,2) + v̺,FE(ωK,2)

6 If ζ ≤ TOL , STOP! 7 For every cell K ∈ TN, and j ∈ {1, 2, · · · , n}, Compute the

Convergence Estimator kK,j.

8 Refine cells according to the modified fraction scheme, ”Dorfler

property”

  • K∈A

k2

K,jkζ2 K ≥ θ2ζ2

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Conclusion

Residual based a posteriori error estimate for conforming hp-AFEM. Validate theoretically and also numerically the reliability and the efficiency of estimator Showing that our hp-adaptive algorithm is a contraction. Introducing a new locally defined reliable and efficient Goal-Oriented a posteriori error estimator for h- and hp- AFEM.

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Future Work

Providing an optimal rate for our Goal-Oriented h-adaptive FEM. Parallelize the code for 3D Stokes problem in Goal-Oriented h-adaptive FEM. Using the above error estimators in both hp- and Goal-Oriented AFEM in the Advanced Solver for Problem in Earth Convection. Parallelizing hp-AFEM?!

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Thank You

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