Linear and Quadratic Finite Volume Methods on Triangular Meshes for Elliptic Equations with Singular Solutions

Abstract

This paper is devoted to the presentation and analysis of some linear and quadratic finite volume (FV) schemes for elliptic problems with singular solutions due to the non-smoothness of the domain. Our FV schemes are constructed over specially-designed graded triangular meshes. We provide sharp parameter selection criteria for the graded mesh, such that both the linear and quadratic FV schemes achieve the optimal convergence rate approximating singular solutions in $H^1$. In addition, we show that on the same mesh, a linear FV scheme obtains the optimal rate of convergence in $L^2$. Numerical tests are provided to verify the analysis.