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Adaptive approximation of elliptic partial differential equations and variational inequalities (joint research with C. Carstensen)

Adaptive mesh refinement has proved to be an efficient tool in the approximation of partial differential equations arising in solid and fluid mechanics. In particular, when singularities or details on small scales have to be resolved, locally refined meshes are advantageous over uniform meshes. The mesh-refinement is steered by local refinement indicators which occur in a posteriori error estimates which have first been proved by Babuska et al. and Verfürth:

Here u is the exact (unknown) solution and u_h is a finite element approximation in a discrete space defined through the triangulation

of the computational domain Omega

. The refinement indicators eta_T

for each triangle T in T

depend on the the approximate solution u_h, the triangulation

, and given right-hand sides. The iterative adaptive approximation of u is then based on the following loop:

Compute the approximate solution u_h on the actual triangulation
Refine those triangles for which to obtain a refined triangulation
Set and go to 1.

Various choices of the tolerance in 2. and of stopping criteria are possible; Dörfler and Morin, Nochetto & Siebert proved convergence of the iteration for one definition of the tolerance in the case of the Poisson equation.

Error estimation by averaging

A very popular but for a long time only heuristically justified choice of the refinement indicators results from an averaging process of u_h: Given the (possibly discontinuous) flux p_h

compute a smooth approximation q_h of p_h and set

for each

. Is it possible to prove an a posteriori error estimate for this choice of eta_T

? The results by B. & Carstensen show that this is indeed the case for any reasonable definition of q_h, there holds

where

is the space of all vector fields with square integrable divergence and

any discrete subspace of H_div

. The design of an approximation operator with additional orthogonality properties is the key to the proof of this estimate. Practical experience shows that the constant C in the error estimate is close to 1. A precise statement of this observation is lacking. Under additional regularity assumptions on the exact solution it can however be shown that the converse, so-called efficiency estimate, holds with constant 1, its proof follows from an application of the triangle inequality,

Figure 1 shows the output of the adaptive approximation scheme in a numerical example in which the exact solution admits large gradients along a circular line. Figure 3 shows that the averaging error estimator serves as a good approximation of the error even on highly unstructured grids as displayed in Figure 2.


Figure 1: Sequence of triangulations generated by an adaptive algorithm	Figure 2: Perturbed adapted triangulation	Figure 3: Convergence rates for uniform, adaptive, and perturbed adaptive approximation

Mixed and non-conforming finite element methods

Mixed and non-conforming finite element methods are important when constraints such as incompressibility are included or when it is necessary to employ parallel solution strategies. The proof of the a posteriori error estimates requires a Helmholtz decomposition of the error and is rather technical in three dimensional settings. Figures 4 and 5 show the numerical solutions for nonconforming and mixed finited element approximation schemes on adaptively refined triangulations in a Poisson model problem.


Figure 4: Non-conforming (discontinuous) finite element approximation	Figure 5: Mixed (piecewise constant) finite element approximation

Higher order methods

It is now well established that higher order finite element methods, the so-called hp-version of the finite element method, may lead to exponentially convergent approximation schemes. This however requires a good balance of a small local mesh-size and a large polynomial approximation degree. Adaptive algorithms turned out to be capable to automatically refine the mesh and increase the polynomial degree locally in such a way that the approximation is exponentially convergent. The global averaging techniques for lowest order methods have to be replaced by sums of local averages in order to obtain a reliable and efficient error estimation. Figures 6, 7, and 8 display the output of higher order approximation schemes and reveal that exponential convergence is possible for adaptive approximation schemes.


Figure 6: Error and estimator for fixed polynomial degree	Figure 7: Error and estimator for adaptive hp-method	Figure 8: Distribution of polynomial degrees on the triangulation

Variational inequalities

Elliptic variational inequalities are more involved as the solutions are typically less regular. In contact problems, the free boundary between the contact zone and the remaining part of the domain has to be resolved by approximation schemes in order to obtain efficient numerical methods. Residual based a posteriori error estimates have first been derived by Nochetto et al. and Veeser. Averaging a posteriori error estimates include an averaging of the gradient of the possibly non-affine obstacle:

Figures 8, 9, and 10 display the result of a numerical experiment. The mesh is automatically refined at the free boundary while the smooth (affine) contact zone remains unrefined. Moreover, as in the linear case, the averaging a posteriori error estimator serves as a good approximation of the actual error.


Figure 9: Numerical approximation - no refinement of the contact zone	Figure 10: Sequence of adaptively refined triangulations and contact zones (in white)	Figure 11: Convergence rates for uniform and adaptive mesh refinement

Animations

The two animations below show the adaptive numerical approximation of elliptic obstacle problems based on averaging error estimators. Click on an icon to download or start the animation (mpeg format).

Related contributions (see articles for further references; downloadable files are preprint versions)

Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis. (With C. Carstensen and G. Dolzmann.) Preprint No. 02-1, Christian Albrechts University Kiel. [.pdf] [Abstract]
Averaging techniques yield reliable a posteriori finite element error control for obstacle problems. (With C. Carstensen.) Preprint No. 01-2, Christian Albrechts University Kiel. [.pdf] [Abstract]
Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM. Math. Comp. 71 No. 239, 945-969 (2002). (With C. Carstensen.) [.pdf] [Abstract]
Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM. Math. Comp. 71 No. 239, 971-994 (2002). (With C. Carstensen.) [.pdf] [Abstract]
An experimental survey of a posteriori Courant finite element error control for the Poisson equation. Adv. Comput. Math. 15 No. 1-4, 79-106 (2001). (With C. Carstensen and R. Klose.) [.pdf] [Abstract]
A posteriori error estimates for nonconforming finite element methods. Numer. Math. 92 No. 2, 233-256 (2002). (With C. Carstensen and S. Jansche.) [.pdf] [Abstract]