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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.