A superconvergent hybridizable discontinuous Galerkin method for the convective Cahn--Hilliard equation

TL;DR

This paper introduces a superconvergent hybridizable discontinuous Galerkin method with convex-concave splitting for the convective Cahn-Hilliard equation, achieving stability, optimal convergence, and efficient solving.

Contribution

The paper develops a novel HDG method with explicit convection discretization, ensuring unconditional stability and superconvergence, along with a specialized projection operator for optimal error estimates.

Findings

Achieves unconditional stability for the convective Cahn-Hilliard equation.

Demonstrates optimal convergence rates in the L2 norm for scalar and flux variables.

Validates theoretical results through numerical experiments showing superconvergence and efficiency.

Abstract

We propose a hybridizable discontinuous Galerkin (HDG) method combined with convex-concave splitting for the temporal discretization of the convective Cahn-Hilliard equation. The convection term is discretized explicitly without stabilization, yielding three key advantages: (1) unconditional stability, (2) preservation of the optimal convergence rate for piecewise constant approximations, and (3) a symmetric system after local elimination, enabling efficient solver via minimal residual methods. We establish optimal convergence rates in the $L^2$ norm for both the scalar and flux variables for any polynomial degree $k \geq 0$. To achieve optimal $L^2$-norm estimates, we introduce a specialized HDG elliptic projection operator and analyze its approximation properties. Within the HDG framework, local elimination is employed to reduce the degrees of freedom associated with the globally coupled unknowns, and the scalar variables exhibit superconvergence. Finally, numerical experiments validate the theoretical convergence rates and demonstrate the effectiveness of the proposed method.