Convergence analysis of a conforming adaptive finite element method for an obstacle problem
- The adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual-based a posteriori error estimator. Each cycle of the adaptive loop consists of the steps 'SOLVE', 'ESTIMATE', 'MARK', and 'REFINE'. The techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of Galerkin orthogonality. We establish R-linear convergence of the part of the energy above its minimal value, if there is appropriate control of the data oscillations. Surprisingly, the adaptive mesh-refinement algorithm is the same as in the unconstrained case of a linear PDE---in fact, there is no modification near the discrete free boundary necessary for R-linear convergence. The arguments are presented for a model obstacle problem with an affine obstacle "chi" and homogeneous Dirichlet boundary conditions. The proof of the discrete local efficiency is more involved than in theThe adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual-based a posteriori error estimator. Each cycle of the adaptive loop consists of the steps 'SOLVE', 'ESTIMATE', 'MARK', and 'REFINE'. The techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of Galerkin orthogonality. We establish R-linear convergence of the part of the energy above its minimal value, if there is appropriate control of the data oscillations. Surprisingly, the adaptive mesh-refinement algorithm is the same as in the unconstrained case of a linear PDE---in fact, there is no modification near the discrete free boundary necessary for R-linear convergence. The arguments are presented for a model obstacle problem with an affine obstacle "chi" and homogeneous Dirichlet boundary conditions. The proof of the discrete local efficiency is more involved than in the unconstrained case. Numerical results are given to illustrate the performance of the error estimator.…
| Author: | Dietrich BraessGND, Carsten CarstensenGND, Ronald H. W. HoppeORCiDGND |
|---|---|
| URN: | urn:nbn:de:bvb:384-opus4-4108 |
| Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/508 |
| Series (Serial Number): | Preprints des Instituts für Mathematik der Universität Augsburg (2007-04) |
| Type: | Preprint |
| Language: | English |
| Date of Publication (online): | 2007/05/29 |
| Publishing Institution: | Universität Augsburg |
| Contributing Corporation: | Department of Mathematics, University of Houston |
| Release Date: | 2007/05/29 |
| Tag: | adaptive finite element method; convergence analysis; obstacle problem |
| GND-Keyword: | Finite-Elemente-Methode; Konvergenz; Hindernisproblem |
| Institutes: | Mathematisch-Naturwissenschaftlich-Technische Fakultät |
| Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik | |
| Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Numerische Mathematik | |
| Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
