An Energy-minimizing Interpolation for Robust Multigrid Methods

We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. The formulation is general; it can be applied to any dimensions. The analysis for one dimension proves that the convergence rate of the resulting multigrid method is independent of the coeecient of the underlying PDE, in addition to being independent of the mesh size. We demonstrate numerically the eeectiveness of the multigrid method in two dimensions by applying it to a discontinuous coeecient problem and an oscillatory coeecient problem. We also show using a one-dimensional Helmholtz problem that the energy minimization principle can be applied to solving elliptic problems that are not positive deenite.