  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 numericaloperators 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.
Efficient implementation of the DG finite element numerical differential
operators is also proposed. 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 DG methods
can be rewritten compactly using the proposed DG finite element differential
calculus framework. Moreover, new DG methods for linear and nonlinear PDEs are
also obtained from the framework.
