 
Summary: STABILITY OF LAGRANGE ELEMENTS FOR THE MIXED
LAPLACIAN
DOUGLAS N. ARNOLD AND MARIE E. ROGNES
Abstract. The stability properties of simple element choices for the mixed
formulation of the Laplacian are investigated numerically. The element choices
studied use vector Lagrange elements, i.e., the space of continuous piecewise
polynomial vector fields of degree at most r, for the vector variable, and the
divergence of this space, which consists of discontinuous piecewise polynomials
of one degree lower, for the scalar variable. For polynomial degrees r equal 2
or 3, this pair of spaces was found to be stable for all mesh families tested. In
particular, it is stable on diagonal mesh families, in contrast to its behaviour
for the Stokes equations. For degree r equal 1, stability holds for some meshes,
but not for others. Additionally, convergence was observed precisely for the
methods that were observed to be stable. However, it seems that optimal order
L2 estimates for the vector variable, known to hold for r > 3, do not hold for
lower degrees.
1. Introduction
In this note, we consider approximations of the mixed Laplace equations with
Dirichlet boundary conditions: Given a source g, find the velocity u and the pressure
p such that
