An Energy-minimizing Interpolation for Robust Multigrid Methods

We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. It can handle PDE coefficients of various types on structured or unstructured grids under one framework. The formulation is general; it can be applied to any dimension. We demonstrate numerically the effectiveness of the multigrid method in two dimensions by applying it to a discontinuous coefficient problem, an oscillatory coefficient problem, and an anisotropic problem. Empirically, the convergence rate is independent of the coefficients of the underlying PDE, in addition to being independent of the mesh size. The proposed method is primarily designed for second-order elliptic PDEs, with possible extensions to other classes of problems such as integral equations.