Uniform convergence of multigrid methods for adaptive meshes

In this paper we study the multigrid methods for adaptively refined finite element meshes. In our multigrid iterations, on each level we only perform relaxation on new nodes and the old nodes whose support of nodal basis function have changed. The convergence analysis of the algorithm is based on the framework of subspace decomposition and subspace correction. In order to decompose the functions from the finest finite element space into each level, a new projection is presented in this paper. Briefly speaking, this new projection can be seemed as the weighted average of the local L2 projection. We can perform our subspace decomposition through this new projection by its localization property. Other properties of this new projection are also presented and by these properties we prove the uniform convergence of the algorithm in both 2D and 3D. We also present some numerical examples to illustrate our conclusion.