Galerkin Difference Approximations: Robust, Efficient, and High-Order Accurate PDE Discretization

Galerkin Difference (GD) methods are a new class of finite element approximations based on a Galerkin projection into a piecewise polynomial space described by a set of known basis functions. However, the derivation of GD basis functions is nonstandard and draws on successful ideas from the finite difference literature. Rather than introducing new interior degrees of freedom, as typical with many FEM methods, GD uses existing degrees of freedom to define high-order accurate basis functions. The effect is to bring together favorable pieces of finite differences and finite elements into a unified framework. In particular, GD discretizations possess an energy estimate guaranteeing stability, are translation invariant away from boundaries which often yields superconvergence at twice the nominal rate, and have favorable stiffness and conditioning properties that permit large CFL=1 time steps.