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Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids

Long Chen, Ricardo H. Nochett, and Chen-Song Zhang

Domain Decomposition Methods in Science and Engineering XIX, 229-236, 2010.

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ABSTRACT: In this paper, we develop and analyze an efficient multigrid method to solve the finite element systems from elliptic obstacle problems on two dimensional adaptive meshes. Adaptive finite element methods (AFEMs) based on local mesh refinement are an important and efficient approach when the solution is non-smooth. An optimality theory on AFEM for linear elliptic equations can be found in Nochetto et al. 2009.

We shall extend the algorithm and theoretical results by Tai 2003 to an important class of adaptive grids obtained by newest vertex bisections; there- after we call them bisection grids for short. This is new according to Graser and Kornhuber 2009: the existing work assumes quasi-uniformity of the un- derlying meshes. Based on a decomposition of bisection grids due to Chen et al. 2009, we present an efficient constraint decomposition method on bi- section grids and prove an almost uniform convergence