A Hybridized Discontinuous Galerkin Method for A Linear Degenerate Elliptic Equation Arising from Two-Phase Mixtures

Abstract

We develop a high-order hybridized discontinuous Galerkin (HDG) method for a linear degenerate elliptic equation arising from a two-phase mixture of mantle convection or glacier dynamics. We show that the proposed HDG method is well-posed by using an energy approach. We derive $apriori$ error estimates for the proposed HDG method on simplicial meshes in both two- and three-dimensions. The error analysis shows that the convergence rates are optimal for both the scaled pressure and the scaled velocity for non-degenerate problems and are sub-optimal by half order for degenerate ones. Several numerical results are presented to confirm the theoretical estimates. We also enhance the HDG solutions by post-processing. The superconvergence rates of $(k+2)$ and $(k+\frac{3}{2})$ are observed for both a non-degenerate case and a degenerate case away from the degeneracy. Degenerate problems with low regularity solutions are also studied, and numerical results show that high-order methods are beneficial in terms of accuracy.

20 pages, 6 figures, 8 tables

References in corpus (2)

• Split Form Nodal Discontinuous Galerkin Schemes with Summation-By-Parts Property for the Compressible Euler Equations
• A hybridizable discontinuous Galerkin method for the Navier--Stokes equations with pointwise divergence-free velocity field

Cited by in corpus (2)

• HDGlab: An open-source implementation of the hybridisable discontinuous Galerkin method in MATLAB
• Unified $hp$-HDG Frameworks for Friedrichs' PDE systems