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A second-order-accurate symmetric discretization of the Poisson equation on irregular domains. (English) Zbl 0996.65108
Summary: 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.

MSC:
65N06 Finite difference methods for boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R35 Free boundary problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
80A22 Stefan problems, phase changes, etc.
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