DR. STANLY STEINBERG
Department of Mathematics and Statistics
University of New Mexico
Albuquerque, New Mexico
stanly@math.unm.edu

Mimetic Discretization of Continuum Mechanics

Problems in continuum mechanics are commonly described by initial boundary value problems for a system of partial differential equations. Such problems can be discretized using finite difference, finite element, spectral, or many related techniques. Mimetic methods follow a different route: they are not used to discretize particular systems of equations, but rather to discretize the continuum theory. Vector calculus provides a powerful invariant (coordinate-free) description of continuum mechanics as does the theory of differential forms. In the vector calculus case, the operators gradient, curl and divergence play a central role: the equation of continuum mechanics can be written in terms of these operators along with the time derivative. Thus mimetic methods for vector calculus provide discretizations of the gradient, curl and divergence, and then these discretizations are used to discretize the partial differential equations that appear in continuum mechanics problems.