Principles of compatible discretizations

Pavel Bochev, Sandia National Lab

Partial differential equations (PDEs) are ubiquitous in science and engineering. A key step in their numerical solution is the discretization that replaces the PDEs by a system of algebraic equations. Like any other model reduction, discretization is accompanied by losses of information about the original problem and its structure. One of the principal tasks in numerical analysis is to develop compatible, or mimetic, algebraic models that yield stable, accurate, and physically consistent approximate solutions. Historically, finite element (FE), finite volume (FV), and finite difference (FD) methods have achieved compatibility by following different paths that reflected their specific approaches to discretization. In spite of their differences, compatible FE, FV, and FD methods can result in discrete problems with remarkably similar properties. The observation that their compatibility is tantamount to having discrete structures that mimic vector calculus identities and theorems emerged independently and at about the same time in the FE, FV, and FD literature. In this talk I will use the classical Kelvin and Dirichlet principles and their associated PDEs to demonstrate the basic principles of compatible (mimetic) discretizations. I use duality to explain why collocated discretizations don't work for mixed Galerkin methods and why such methods require staggered or dual-primal grids. Then I present three basic types of compatible methods for the model equations that give rise to mixed Galerkin, co-volume and least-squares type methods. The rigorous mathematics behind spatial compatibility, including De Rham differential complex and co-homology, exactness and applications of algebraic topology to derive mimetic methods is the subject of the main talk.