M-Adaptation in the mimetic finite difference method

The mimetic finite difference method produces a family of schemes with equivalent properties such as the stencil size, stability region, and convergence order. Each member of this family is defined by a set of parameters which can be chosen locally for every mesh element. The number of parameters depends on the geometry of a particular mesh element. M-Adaptation is a new adaptation methodology that identifies a member of this family with additional (superior) properties compared to the other schemes in the family. We analyze the enforcement of the discrete maximum principles for the diffusion equation in the primal and dual forms, the reduction of numerical dispersion and anisotropy for the acoustic wave equation, and the optimization of the performance of multi-grid solvers.