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This paper develops a discontinuous Galerkin (DG) finite element differentialcalculus theory for approximating weak derivatives of Sobolev functions andpiecewise Sobolev functions. By introducing numerical one-sided derivatives asbuilding blocks, various first and second order numericaloperators such as thegradient, divergence, Hessian, and Laplacian operator are defined, and theircorresponding calculus rules are established. Among the calculus rules areproduct and chain rules, integration by parts formulas and the divergencetheorem. Approximation properties and the relationship between the proposed DGfinite element numerical derivatives and some well-known finite differencenumerical derivative formulas on Cartesian grids are also established.Efficient implementation of the DG finite element numerical differentialoperators is also proposed. Besides independent interest in numericaldifferentiation, the primary motivation and goal of developing the DG finiteelement differential calculus is to solve partial differential equations. It isshown that several existing finite element, finite difference and DG methodscan be rewritten compactly using the proposed DG finite element differentialcalculus framework. Moreover, new DG methods for linear and nonlinear PDEs arealso obtained from the framework.

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