On the superconvergence of a hydridizable discontinuous Galerkin method for the Cahn-Hilliard equation

TL;DR

We present an HDG method for the fourth order nonlinear Cahn-Hilliard equation with temporal discretization options of backward Euler or convex splitting. The paper proves existence, uniqueness and energy stability for the fully discrete scheme, and establishes optimal L^2 convergence for all variables, with superconvergence of the scalar traces and a novel HDG Sobolev inequality to handle nonlinear analysis. An HDG elliptic projection is constructed to derive sharp error bounds, including a negative-norm estimate for the scalar variable when k >= 1. Numerical experiments on unit squares confirm the theoretical rates and the anticipated superconvergence, highlighting the method's effectiveness for fourth-order nonlinear PDEs with reduced globally coupled degrees of freedom.

Abstract

We propose a hydridizable discontinuous Galerkin (HDG) method for solving the Cahn-Hilliard equation. The temporal discretization can be based on either the backward Euler method or the convex-splitting method. We show that the fully discrete scheme admits a unique solution, and we establish optimal convergence rates for all variables in the $L^2$ norm for arbitrary polynomial orders. In terms of the globally coupled degrees of freedom, the scalar variables are superconvergent. Another theoretical contribution of this work is a novel HDG Sobolev inequality that is useful for HDG error analysis of nonlinear problems. Numerical results are reported to confirm the theoretical convergence rates.

Theorems & Definitions (70)

• Lemma 3.1
• Lemma 3.2: The HDG Poincaré inequality
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• proof
• Lemma 3.6
• proof
• Lemma 3.7: Discrete LBB Condition of $\mathcal{A}$
• proof