Curvilinear elements and nonconforming discretizations

All previous chapters have focused almost exclusively on the simplest cases of geometric discretizations where all boundaries are assumed to be piecewise linear and all elements share faces of equal size and order of approximation. However, one of the major advantages of discontinuous Galerkin (DG) methods lies in their flexibility to go beyond these cases and support more complex situation.

In this chapter we discuss the modifications required to extend the linear conforming elements to include the treatment of meshes containing curvilinear elements and/or non-conforming elements. As we will see, the required changes are limited, but the advantages of doing so can be dramatic in terms of improvements in accuracy and reductions in computational effort.