Multigrid Methods for Elliptic Obstacle Problems on 2D Bisection Grids

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]. To achieve optimal complexity, an efficient solver for the discretization is indispensable. The classical projected successive over-relaxation method by Cryer [1979] converges but the convergence rate degenerates quickly as the mesh size approaches zero. To speed up the convergence, different multigrid and domain decomposition techniques have been developed (see the monograph by Kornhuber [1997] and the recent review by Graser and Kornhuber [2009].) In particular, the constraint decomposition method by Tai [2003] is proved to be convergent linearly with a rate which is almost robust with respect to the mesh size in R; but the result is restricted to uniformly refined grids. We shall extend the algorithm and theoretical results by Tai [2003] to an important class of adaptive grids obtained by newest vertex bisections; thereafter we call them bisection grids for short. This is new according to Graser and Kornhuber [2009]: the existing work assumes quasi-uniformity of the underlying meshes. Based on a decomposition of bisection grids due to Chen et al. [2009], we present an efficient constraint decomposition method on bisection grids and prove an almost uniform convergence