Stability of Lagrange elements for the mixed Laplacian

TL;DR

This paper investigates the stability of simple vector Lagrange element pairs for the mixed Laplacian problem, revealing stability for degrees 2 and 3 across various meshes and highlighting issues at degree 1.

Contribution

It provides numerical evidence on the stability of vector Lagrange elements for the mixed Laplacian, especially for degrees 2 and 3, and discusses stability variations at degree 1.

Findings

Stable for degrees 2 and 3 across tested meshes.

Unstable at degree 1 for some mesh families.

Observed convergence only for stable methods.

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.