CRPC-TR95525-S:  Implementation of Mixed Finite Element Methods for 
Elliptic Equations on General Geometry

Todd Arbogast, Clint N. Dawson, Philip T. Keenan, Mary F. Wheeler, Ivan 
Yotov 

March, 1995

We consider the efficient implementation of mixed finite elements for
solving second order elliptic partial differential equations on
geometrically general domains, concentrating on the lowest-order
Raviart-Thomas approximating spaces.  We consider the standard mixed
method and its hybrid form, and the recently introduced expanded mixed
method.  The standard method yields a saddle-point linear system, and
while the hybrid method yields a positive definite linear system, it
has many more unknowns, one per element edge or face.  The expanded
mixed method is similar in its structure; however, we give a
generalization of the method combined with a global mapping technique
that makes it suitable for general meshes.  Moreover, two quadrature
rules are given which reduce the method to a cell-centered finite
difference method on meshes of quadrilaterals or triangles in 2
dimensions and hexahedra or tetrahedra in 3 dimensions.  This approach
substantially reduces the complexity of the mixed finite element
matrix, leaving a symmetric, positive definite system for only as many
unknowns as elements.  On smooth meshes that are either logically
rectangular or triangular with six triangles per internal vertex, this
finite difference method is as accurate as the standard mixed method;
on non-smooth meshes it can lose accuracy.  An enhancement of the
method is defined that combines numerical quadrature with Lagrange
multiplier pressures on certain element edges or faces.  The enhanced
method regains the accuracy of the solution on non-smooth meshes, with
little additional cost if the mesh geometry is piece-wise smooth, as
in hierarchical meshes.  Theoretical error estimates and numerical
examples are given comparing the accuracy and efficiency of the methods.