A superconvergent hybridizable discontinuous Galerkin method for the convective Cahn--Hilliard equation
This work provides a stable, optimally convergent numerical method for the convective Cahn-Hilliard equation, which is important for phase-field simulations in fluid dynamics.
The paper proposes an HDG method with convex-concave splitting for the convective Cahn-Hilliard equation, achieving unconditional stability and optimal convergence rates for any polynomial degree k≥0. Numerical experiments confirm the theoretical rates.
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.