Recovery-Based a Posteriori Error Estimators for Interface Problems: Mixed and Nonconforming Elements

Transcription

1 Recovery-Based a Posteriori Error Estimators for Interface Problems: Mixed and Nonconforming Elements Zhiqiang Cai Shun Zhang Department of Mathematics Purdue University Finite Element Circus, Fall 2008, RPI

2 Outline Introductions A Model Problem Guidelines of Recovery-Based Error Estimators Interface Problems Mixed FEM for Interface Problems Nonconforming FEM for Interface Problems Conclusions

3 A Priori Error Estimation Continuous Problem Lu = f in Ω Discrete Problem L h u h = f h a Priori Error Estimation u u h C(u)h α 0 as h 0 Error Control ɛ u u h ɛ

4 A Posteriori Error Estimations A Posteriori Error Estimations Indicator ηk (u h ) - a computable quantity for each K T Estimator η(u h ) = ( K T η2 K Error Control: Reliability Bound ) 1/2 u u h C r η + h.o.t Adaptive Control of Meshing: Efficiency Bound η K c e u u h K + h.o.t K T η C e u u h + h.o.t

5 A Posteriori Error Estimations Robustness C r and C e are independent of the parameters inherent in the differential equations, such as jumps of the diffusion coefficients, reaction/convetion parameters, e.t.c Effectivity Constant η u u h If the effectivity constant is close to 1, the estimator is accurate.

6 Recovery-Based Estimators Recovery-based Estimators: σ(u h ) is a quantity of mathematical or physical meaning, such as gradient, flux or stress, recover it to get G(σ(u h )) in an appropriate function space with an appropriate norm (most time the energy norm from the differential equation) η G = G(σ(u h )) σ(u h ) Possible Good Quality of Recovery-based Estimators: η Effectivity constant is close to 1 u u h

7 An Analysis of a Model Problem Diffusion Equations (A u) = f Ω Smoothness of the Problem Solution: u H 1 (Ω) Continuous (in the weak sense) Gradient: u H(curl; Ω) Tangential Component is Continuous Flux: σ = A u H(div; Ω) Normal Component is Continuous

8 Guidelines of Recovery-Based Error Estimators Recover a quantity whose continuity is violated by the discretization method (Conforming FEM, Mixed Methods, Nonconforming Element Methods...) In the corresponding conforming finite element space( without introducing unnecessary continuity), Measure the difference in the right norm (Energy Norm) as the indicator.

9 Interface Problems (a(x) u) = f in Ω R d u = 0 on Γ D a u n = 0 on Γ N a(x) is positive piecewise constant w.r.t Ω = n i=1 Ω i, a(x) = a i > 0 in Ω i

10 Low Order Mixed FEM for Interface Problems The corresponding mixed variational formulation is to find (σ, u) H N (div; Ω) L 2 (Ω) such that { (a 1 σ, τ ) ( τ, u) = 0 τ H N (div; Ω), ( σ, v) = (f, v) v L 2 (Ω). Discrete Problem: The mixed finite element method is to find (σ m, u m ) RT 0 P 0 such that { (a 1 σ m, τ ) ( τ, u m ) = 0 τ RT 0, ( σ m, v) = (f, v) v P 0.

11 Mixed FEM for Interface Problems Comparison of Continuous and Discrete Solutions: Solution u H 1 D (Ω) u h P 0 H 1 D (Ω) Gradient u H(curl; Ω) a 1 σ m H(curl; Ω) Flux σ = a u H(div; Ω) σ m RT 0 H(div; Ω) Quantity to recover: the Gradient (from a 1 σ m ). In what space? H(curl; Ω)-conforming element spaces ND (Nedlec edge element spaces of type 1 and 2) What if recover in global continuous space S 2 1? Will introduce unnecessary over refinements! S 1 = {v : v C 0 (Ω), v K P 1 (K ), K T }

12 Robust Gradient Recovery Error Estimators for Interface Problems: Mixed FEs L 2 -Projection Gradient Recovery: Find ρ m ND 2 such that (a ρ m, τ ) = (σ m, τ ) τ ND 2. Explicit Recovery: See Cai & Zhang 08 for details. Error Estimator: η m,k = a 1/2 ρ m +a 1/2 σ m 0,K, η m = a 1/2 ρ m +a 1/2 σ m 0,Ω,

13 Robust Gradient Recovery Error Estimators for Interface Problems: Mixed FEs Robustness: C e and C r is independent of the jumps of the coefficients across the interfaces Ce 1 η m + h.o.t a 1/2 u + a 1/2 σ m 0,Ω C r η m + h.o.t Accurateness: Observed in numerical tests that the effectivity index is close to 1. No over refinements along the interface!

14 A Benchmark Test Ptoblem interface problem { (a u) = f in Ω = ( 1, 1) 2 u = g on Ω with a = R in (0, 1) 2 ( 1, 0) 2 and 1 in ( 1, 0) (0, 1) (0, 1) ( 1, 0) exact solution u(r, θ) = r α µ(θ) H 1+α ɛ (Ω) with cos(( π 2 σ)α) cos((θ π 2 + ρ)α) if 0 θ π 2, π cos(ρα) cos((θ π + σ)α) if µ(θ) = 2 θ π, cos(σα) cos((θ π ρ)α) if π θ 3π 2, cos(( π 2 ρ)α) cos((θ 3π 2 + σ)α) if 3π 2 θ 2π. example α = 0.1 u H 1.1 ɛ (Ω) R , ρ = π/4, and σ

15 Numerical Results by Robust Gradient Recovery Error Estimators: Mixed FEs η m,rt / k 1/2 u 0 and k 1/2 u+k 1/2 σ m 0 / k 1/2 u η m,rt / k 1/2 u 0 k 1/2 u+k 1/2 σ m 0 / k 1/2 u reference line with slope 1/ number of nodes Figure: mesh generated by η m Figure: error and η m

16 Gradient/Flux Recovery Error Estimators in Continuous S 2 1 Let σ m be the solution, and let ρ m,f S 2 1 and ρ m,g S 2 1 satisfy the following problems (a 1 ρ m,f, τ ) = (a 1 σ m, τ ) τ S 2 1 and (aρ m,g, τ ) = (σ m, τ ) τ S 2 1, respectively. Then the corresponding error estimators are defined by η m,cb,f = a 1/2 (σ m ρ m,f ) 0,Ω η m,cb,g = a 1/2 σ m + a 1/2 ρ m,g 0,Ω.

17 Gradient/Flux Recovery Error Estimators in Continuous S 2 1 Lots of over refinements along the interface because of recovering in a space asking too much continuity! Figure: mesh by η m,cb,f Figure: mesh by η m,cb,g

18 Linear Nonconforming FEM for Interface Problems Variational Problem: To find u HD 1 (Ω), such that (a u, v) = (f, v) v H 1 D (Ω) Linear Nonconforming FE Space(Crouzeix-Raviart): V nc = {v L 2 (Ω) : v K P 1 (K ) K T, and v is continuous at m e e E Ω, v = 0 onγ D } Discrete Problem: The nonconforming finite element method is to find u nc V nc such that (a h u nc, h v nc ) = (f, v nc ) v nc V nc

19 Nonconforming FEM for Interface Problems Comparison of Continuous and Discrete Solutions: Solution u H 1 D (Ω) u h V nc H 1 D (Ω) Gradient u H(curl; Ω) h u nc H(curl; Ω) Flux σ = a u H(div; Ω) a h u nc H(div; Ω) Quantities to recover: Both the Gradient (from h u nc ) and the Flux (from a h u nc ). In what spaces? the Gradient in H(curl; Ω)-conforming element spaces ND and the Flux in H(div; Ω)-conforming element spaces What if in S1 2? Will introduce unnecessary refinements!

20 Robust Gradient/Gradient Recovery Error Estimators for Interface Problems: Nonconforming FEs L 2 -Projection Gradient Recovery: Find ρ nc ND 1 such that (a ρ nc, τ ) = (a h u nc, τ ) τ ND 1. L 2 -Projection Flux Recovery: Find σ nc RT 0 such that (a 1 σ nc, τ ) = ( h u nc, τ ) τ RT 0. Explicit Recovery: See Cai & Zhang 08 for details. Error Estimator: with η 2 nc = c 2 η 2 nc,1 + (1 c2 )η 2 nc,2 for 0 < c < 1. η nc,1 = a 1/2 σ nc + a 1/2 h u nc 0,Ω η nc,2 = a 1/2 (ρ nc + h u nc ) 0,Ω

21 Robust Gradient Recovery Error Estimators for Interface Problems: Mixed FEs Robustness: C e and C r is independent of the jumps of the coefficients across the interfaces Ce 1 η nc + h.o.t a 1/2 u a 1/2 h u nc 0,Ω C r η nc + h.o.t Accurateness: Observed in numerical tests that the effectivity constant is close to 1. No over refinements along the interface!

22 Numerical Results by Robust Gradient/Flux Recovery Error Estimators: Nonconforming FEs The test problem is the same interface problem introduced before. η nc / k 1/2 u 0 and k 1/2 ( u h u nc ) 0 / k 1/2 u η nc / k 1/2 u 0 k 1/2 ( u h u nc ) 0 / k 1/2 u reference line with slope 1/ number of nodes Figure: mesh generated by η nc Figure: error and η nc

23 Gradient/Flux Recovery Error Estimators in Continuous S1 2 for Nonconforming FEs let ρ nc,f S 2 1 and ρ nc,g S 2 1 satisfy the following problems (a 1 ρ nc,f, τ ) = ( h u nc, τ ) τ S 2 1 and (aρ nc,g, τ ) = (a h u nc, τ ) τ S 2 1. Then the corresponding error estimators are defined by η nc,cb,f = a 1/2 h u nc + a 1/2 ρ nc,f 0,Ω η nc,cb,g = a 1/2 ( h u nc ρ nc,g ) 0,Ω.

24 Gradient/Flux Recovery Error Estimators in Continuous S1 2 for Nonconforming FEs Lots of over refinements along the interface because of recovering in a space aking for too much continuity! Figure: mesh by η m,cb,f Figure: mesh by η m,cb,g

25 Conclusions Guidelines for choose quantities, spaces in recovery based a posteriori error estimators Robust recovery error estimators for lower order mixed and nonconforming elements