A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems

TL;DR

This paper introduces a novel mixed discontinuous Galerkin method for time-dependent fourth order problems that achieves stability and optimal error estimates without interior penalty, demonstrated through numerical experiments and an application to the Swift-Hohenberg equation.

Contribution

A new mixed DG method for fourth order PDEs that is stable without interior penalty and provides optimal error estimates, with extensions and practical applications.

Findings

Method is unconditionally stable and second order in time.

Optimal $L^2$ error estimate of $O(h^{k+1})$ for semi-discrete schemes.

Numerical results confirm stability and accuracy.

Abstract

A novel discontinuous Galerkin (DG) method is developed to solve time-dependent bi-harmonic type equations involving fourth derivatives in one and multiple space dimensions. We present the spatial DG discretization based on a mixed formulation and central interface numerical fluxes so that the resulting semi-discrete schemes are $L^2$ stable even without interior penalty. For time discretization, we use Crank-Nicolson so that the resulting scheme is unconditionally stable and second order in time. We present the optimal $L^2$ error estimate of $O(h^{k+1})$ for polynomials of degree $k$ for semi-discrete DG schemes, and the $L^2$ error of $O(h^{k+1} +(\Delta t)^2)$ for fully discrete DG schemes. Extensions to more general fourth order partial differential equations and cases with non-homogeneous boundary conditions are provided. Numerical results are presented to verify the stability and accuracy of the schemes. Finally, an application to the one-dimensional Swift-Hohenberg equation endowed with a decay free energy is presented.