Adaptive mesh refinement for Multigrid Solver

In this work, we introduce the Galerkin finite element Method for Elliptic Problems. The estimates of the approximation error in both energy norm and $L2$ are given for the variational formulation of the Poisson problem, discretized by the Galerkin finite element method. Then, the adaptive mesh refinement from Quarteroni is applied to solve the multigrid Poisson problem. This refinement is proven to be very efficient and effective compared with the uniform mesh refinement. Moreover, we propose a new estimator for the adaptive mesh refinement based on the error of approximate solution in the adjacent levels. The numerical results show that the adaptive mesh refinement with a new estimator performs much better than the one with an estimator from Quarteroni in terms of both computational time and the number of elements.