A second order virtual node method for elliptic problems with interfaces and irregular domains

  • Authors:

  • Jacob Bedrossian;James H. von Brecht;Siwei Zhu;Eftychios Sifakis;Joseph M. Teran

  • Affiliations:

  • University of California-Los Angeles, Department of Mathematics, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095, United States;University of California-Los Angeles, Department of Mathematics, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095, United States;University of California-Los Angeles, Department of Mathematics, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095, United States;University of California-Los Angeles, Department of Mathematics, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095, United States;University of California-Los Angeles, Department of Mathematics, 520 Portola Plaza, Math Sciences Building 6363, Los Angeles, CA 90095, United States

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2010

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Abstract

We present a second order accurate, geometrically flexible and easy to implement method for solving the variable coefficient Poisson equation with interfacial discontinuities or on irregular domains, handling both cases with the same approach. We discretize the equations using an embedded approach on a uniform Cartesian grid employing virtual nodes at interfaces and boundaries. A variational method is used to define numerical stencils near these special virtual nodes and a Lagrange multiplier approach is used to enforce jump conditions and Dirichlet boundary conditions. Our combination of these two aspects yields a symmetric positive definite discretization. In the general case, we obtain the standard 5-point stencil away from the interface. For the specific case of interface problems with continuous coefficients, we present a discontinuity removal technique that admits use of the standard 5-point finite difference stencil everywhere in the domain. Numerical experiments indicate second order accuracy in L^~.