Stability of finite volume approximations for the Laplacian operator on quadrilateral and triangular grids
The objective of this paper is to analyse an edge based finite volume approximation of the Laplacian and construct an accurate and stable scheme to implement boundary conditions for time dependent problems. Of particular interest are unstructured grids where the finite volume method is fully advantageous. The heat equation is taken as a model problem. The Cauchy problem in one and several space dimensions is analyzed and the stability on unstructured grids is proven by showing the negative semidefiniteness of the matrices. Next, the initial-boundary value problem is considered and a scheme is constructed in a summation by parts framework. The boundary conditions are imposed in a stable and accurate manner using a penalty formulation. Numerical computations are performed for the wave equation in two-dimensions, verifying stability and order of accuracy for structured grids. However, the results are not satisfactory for unstructured grids, and further investigation reveals that the approximation is not consistent for general unstructured grids. However, grids consisting of equilateral polygons recover the convergence. It appears that the required stability analysis should be different from that for the heat equation.
scientific article; zbMATH DE number 2107330
scientific article