Discretization schemes on triangular grids Original Research Article

Physical applications posed on irregular domains have caused difficulties in the use of many, otherwise effective, discretization schemes. In two-dimensional examples, we consider discretization schemes on triangles to approximate irregular domains. Although finite element, mixed finite element, and finite volume schemes for triangles are well understood, efficient solvers are essential for their effective use. In this paper, we consider certain cell-centered finite difference schemes on triangles, both for their theoretical properties on regular grids, and as effective preconditioners for other discretization schemes for more general triangulations. Two cell-centered finite difference schemes (four-point star stencils) are derived from the simplification of the lowest-order Raviart-Thomas mixed finite element discretization on certain triangular grids. Optimal and superconvergence error estimates for both the solution and the gradient of the solution are established. The accuracy of these schemes is also discussed in the case of discontinuous coefficients. Computational experiments verify the error estimates. Finally, the use of these methods as preconditioners is also discussed.