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

TL;DR

This paper introduces a high-order hybridized discontinuous Galerkin method for a linear degenerate elliptic equation from two-phase mixtures, providing theoretical error estimates and numerical validation for both non-degenerate and degenerate cases.

Contribution

The paper develops a novel HDG method with proven well-posedness, optimal convergence rates for non-degenerate problems, and superconvergence enhancements, specifically addressing degenerate elliptic equations.

Findings

Optimal convergence for non-degenerate problems

Sub-optimal convergence for degenerate problems

Superconvergence rates observed in numerical tests

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 ${\it a priori}$ 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.