LBB Stability of a Mixed Discontinuous/Continuous Galerkin Finite Element Pair
Provides a new flexible element choice for triangular/tetrahedral meshes that ensures LBB stability, addressing a known bottleneck in mixed finite element methods.
Introduced a new mixed discontinuous/continuous Galerkin finite element (P1dg-P2) for wave and incompressible flow equations that satisfies the LBB stability condition, eliminating spurious zero-energy modes. Demonstrated stability through numerical tests and analyses in 1D, 2D, and 3D.
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.