Stability of lagrange elements for the mixed Laplacian

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 behavior 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.