A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives

Abstract

Abstract. In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal (k + 1)-th order of accuracy when using piecewise k-th degree polynomials, under the condition that k + 1 is greater than or equal to the order of the equation.

Polymorphic nodal elements and their application in discontinuous Galerkin methods

“…This is particularly true for viscous problems, where the common solution approach is based on a mixed finite element formulation, which was introduced in [3] and extended to higher order problems in [38,39]. In recent developments for the DG discretization of second order terms [14,15,30], the introduction of auxiliary variables is circumvented by the use of two partial integrations, or by multiple partial integrations for higher order operators [6].…”

A Finite Element Model Based on Discontinuous Galerkin Methods on Moving Grids for Vertebrate Limb Pattern Formation

“…As in [29], we formally apply the DG formulation [3] to discretize the reaction-diffusion equations (2.2) in the spatial dimensions, but keep the time variable continuous. The difference from [29] is that now the finite element space is time-dependent since we are solving the problem on a moving domain.…”

“…Following [3], we take β = 10/h min (t). The choice of numerical fluxes (2.5)-(2.7) is crucial for the stability and convergence of the DG scheme (2.4).…”

“…The choice of numerical fluxes (2.5)-(2.7) is crucial for the stability and convergence of the DG scheme (2.4). See [6,3] for further discussion of the choice of numerical fluxes.…”

“…The method is based on a new DG method for solving time dependent PDEs with higher order spatial derivatives, developed by Cheng and Shu [3]. These investigators formulated the scheme by repeated integration by parts of the original equation and then replacing the interface values of the solution by carefully chosen numerical fluxes.…”