Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations

Abstract

This paper develops a discontinuous Galerkin (DG) finite element differential calculus theory for approximating weak derivatives of Sobolev functions and piecewise Sobolev functions. By introducing numerical one-sided derivatives as building blocks, various first and second order numerical operators such as the gradient, divergence, Hessian, and Laplacian operator are defined, and their corresponding calculus rules are established. Among the calculus rules are product and chain rules, integration by parts formulas and the divergence theorem. Approximation properties and the relationship between the proposed DG finite element numerical derivatives and some well-known finite difference numerical derivative formulas on Cartesian grids are also established. Besides independent interest in numerical differentiation, the primary motivation and goal of developing the DG finite element differential calculus is to solve partial differential equations. It is shown that several existing finite element, finite difference and discontinuous Galerkin methods can be rewritten compactly using the proposed DG finite element differential calculus framework. Moreover, new discontinuous Galerkin methods for linear and nonlinear PDEs are also obtained from the framework.