Department of Applied Mathematics

University of Colorado at Boulder

Campus Box 526

Boulder, Colorado 80309-0526

Tom Manteuffel, Steve McCormick, Luke Olson

Abstract

Least-squares finite element methods for the inviscid Burgers
equation in space-time domains are presented. Numerical
results show that convergence of a standard least-squares approach
for div(*u*,*u*^{2}/2)=0
with bilinear elements for *u* and
Newton linearization is problematic, possibly due to difficulties
with the linearizability of the Burgers operator around discontinuous
solutions. Alternative H(div)-conforming formulations are proposed in
terms of the flux variables (or their De Rahm-dual) using face and
edge elements. The face element formulation does not exhibit exact
numerical conservation at the discrete level, but it is
shown that the formulation converges to a conservative weak
solution with the correct shock speed. The dual edge element
formulation is strictly conservative at the discrete level.
The least-squares functional naturally provides a sharp a posteriori
error estimator that is used for adaptive refinement in space-time.
Standard algebraic multigrid methods are applied to the positive
semi-definite matrices resulting from the new least-squares
formulations, and multigrid efficiency as a function of
problem size is investigated.
Extension of the new least-squares methods to general
systems of hyperbolic conservation laws in multiple dimensions
is discussed.