A Second-Order-Accurate Symmetric Discretization of the Poisson Equation on Irregular Domains

In this paper, we consider the variable coefficient Poisson equation with Dirichlet boundary conditions on an irregular domain and show that one can obtain second-order accuracy with a rather simple discretization. Moreover, since our discretization matrix is symmetric, it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second-order-accurate discretizations of this problem. Multidimensional computational results are presented to demonstrate the second-order accuracy of this numerical method. In addition, we use our approach to formulate a second-order-accurate symmetric implicit time discretization of the heat equation on irregular domains. Then we briefly consider Stefan problems.