Shape Regularity Conditions for Polygonal/Polyhedral Meshes, Exemplified in a Discontinuous Galerkin Discretization

This article concerns shape regularity conditions on arbitrarily shaped polygonal/polyhedral meshes. In (J. Wang and X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems, Math Comp 83 (2014), 2101-2126), a set of shape regularity conditions has been proposed, which allows one to prove important inequalities such as the trace inequality, the inverse inequality, and the approximation property of the L-2 projection on general polygonal/polyhedral meshes. In this article, we propose a simplified set of conditions which provides similar mesh properties. Our set of conditions has two advantages. First, it allows the existence of small edges/faces, as long as the shape of the polygon/polyhedron is regular. Second, coupled with an extra condition, we are now able to deal with nonquasiuniform meshes. As an example, we show that the discontinuous Galerkin method for Laplacian equations on arbitrarily shaped polygonal/polyhedral meshes, satisfying the proposed set of shape regularity conditions, achieves optimal rate of convergence. (c) 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 308-325, 2015