A Semi-Uniform Multigrid Algorithm for Solving Elliptic Interface Problems

Abstract

Abstract We introduce a new geometric multigrid algorithm to solve elliptic interface problems. First we discretize the problems by the usual P 1 {P_{1}} -conforming finite element methods on a semi-uniform grid which is obtained by refining a uniform grid. To solve the algebraic system, we adopt subspace correction methods for which we use uniform grids as the auxiliary spaces. To enhance the efficiency of the algorithms, we define a new transfer operator between a uniform grid and a semi-uniform grid so that the transferred functions satisfy the flux continuity along the interface. In the auxiliary space, the system is solved by the usual multigrid algorithm with a similarly modified prolongation operator. We show 𝒲 {\mathcal{W}} -cycle convergence for the proposed multigrid algorithm. We demonstrate the performance of our multigrid algorithm for problems having various ratios of parameters. We observe that the computational complexity of our algorithms are robust for all problems we tested.