Discontinuous Galerkin finite element methods (DG FEM) for the shallow water equations using mixed element meshes that consist of triangular and quadrilateral elements in two-dimensions and triangular prisms and hexahedra in three-dimensions are developed, implemented, and tested. The main motivation behind this work is to gain more insight on whether the use of quadrilateral/hexahedral elements improves the efficiency of DG methods in a setting in which two (adjacent) triangular/triangular prism elements are merged to form a single quadrilateral/hexahedral element. The elements that are used in this study are constructed from a set of orthogonal, modal basis functions formed from products of Legendre and Jacobi polynomials. Given the fact that DG methods do not require continuity of the approximate solution between elements, quadrilateral and hexahedral element basis functions may be developed that exclude the usual cross-terms that are present in standard C0 elements, e.g., a linear quadrilateral element may be used instead of a bilinear quadrilateral element. The performance of the developed DG methods on triangular meshes and quadrilateral meshes of arbitrary polynomial order p is evaluated in terms of accuracy and computational time on a set of analytic test cases for the linear shallow water equations. The numerical results provide evidence that there is substantial benefit in using quadrilateral elements, and it is expected that the hexahedral elements will offer similar computational savings over the triangular prism elements.
This work also focuses on improving the computational efficiency of the existing triangular prism elements. FEM is rapidly progressing in multi-dimensional and multi-regional domains in which a crucial point in computing the FEM solutions is the evaluation of the domain integrals arising over the master element. Previously, nonproduct rules have not been developed specifically for the triangular prism, as they have for many other commonly used shapes. Shape--specific nonproduct rules for numerical integration over the triangular prism domain are derived in this work, which result in, on average, a 27% computational savings over each element for each time step of the problem.
This research has laid the groundwork for a robust computational infrastructure in the context of DG FEM that will significantly cut computational cost by applying fast and efficient state-of-the-art algorithms. The promising results provide motivation for future model development within the framework of a mixed element approach as well as the derivation of higher order numerical integration rules for triangular prism domains.