LBB Stability of a Mixed Discontinuous/Continuous Galerkin Finite Element Pair

TL;DR

This paper introduces a new mixed finite element method combining discontinuous and continuous functions for wave and flow equations, ensuring stability and eliminating spurious modes on complex meshes.

Contribution

A novel P1dg-P2 mixed element that satisfies LBB stability for triangular and tetrahedral meshes, enhancing flexibility and accuracy in wave and flow simulations.

Findings

The P1dg-P2 element is LBB stable for 2D and 3D problems.

Numerical tests confirm the absence of spurious zero-energy modes.

Analysis shows improved stability properties over existing methods.

Abstract

We introduce a new mixed discontinuous/continuous Galerkin finite element for solving the 2- and 3-dimensional wave equations and equations of incompressible flow. The element, which we refer to as P1dg-P2, uses discontinuous piecewise linear functions for velocity and continuous piecewise quadratic functions for pressure. The aim of introducing the mixed formulation is to produce a new flexible element choice for triangular and tetrahedral meshes which satisfies the LBB stability condition and hence has no spurious zero-energy modes. We illustrate this property with numerical integrations of the wave equation in two dimensions, an analysis of the resultant discrete Laplace operator in two and three dimensions, and a normal mode analysis of the semi-discrete wave equation in one dimension.