Subject: a third tr abstract

From: [email-masked] (Todd Arbogast)
Date: Tue, 14 Jun 94 10:36:51 CDT

MIXED FINITE ELEMENTS AS FINITE DIFFERENCES FOR
ELLIPTIC EQUATIONS ON TRIANGULAR ELEMENTS

Todd Arbogast, Clint N. Dawson, and Philip T. Keenan

Abstract:  Five procedures of mixed finite element type for
solving elliptic partial differential equations on triangular meshes
are presented: the standard and hybrid mixed methods, the recently
introduced expanded mixed method, and two new methods.  The efficient
implementation of these procedures using the lowest-order
Raviart-Thomas approximating spaces defined on triangular elements is
discussed. The standard method yields a saddle-point linear system,
and while the hybrid method yields a positive definite linear system,
it uses 50% more unknowns. A quadrature rule is given which reduces a
new, expanded formulation of the mixed method to a finite difference
method on triangles. This approach substantially reduces the
complexity of the mixed finite element matrix. On smooth meshes this
new approach appears to be as accurate as the standard method; on
non-smooth meshes it can lose accuracy. An enhancement of this method
is derived which combines numerical quadrature with Lagrange
multipliers on certain element edges.  The enhanced method regains the
accuracy of the solution, with little additional cost if the mesh
geometry is piece-wise smooth, as in hierarchical meshes.  Numerical
examples in two dimensions are given comparing the accuracy of the
methods.

Keywords:  Mixed finite element method, elliptic partial
differential equation, finite differences, triangles