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

JOURNAL OF COMPUTATIONAL PHYSICS

卷:231

Article

Hellrung, Jeffrey Lee, Jr.¹  Wang, Luming²  Sifakis, Eftychios³  Teran, Joseph M.¹ 

[1] Univ Calif Los Angeles, Dept Math, Los Angeles, CA 90095 USA

[2] Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA

[3] Univ Wisconsin, Dept Comp Sci, Madison, WI 53706 USA

关键词: Elliptic interface problems;    Embedded interface methods;    Virtual node methods;    Variational methods;    Multigrid methods;   

DOI  :  10.1016/j.jcp.2011.11.023

来源: Elsevier

【 摘 要 】

We present a numerical method for the variable coefficient Poisson equation in three-dimensional irregular domains and with interfacial discontinuities. The discretization embeds the domain and interface into a uniform Cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions. The matrix associated with the discretization is symmetric positive definite and equal to the standard 7-point finite difference Poisson stencil away from embedded interfaces and boundaries. Numerical evidence suggests second order accuracy in the L-infinity'-norm. Our approach improves the treatment of Dirichlet and jump constraints in the recent work of Bedrossian et al. [1] and introduces innovations necessary in three dimensions. Specifically, we construct new constraint-based Lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations; we provide a method for cell-local polyhedral approximation to the zero isocontour surface of a level set needed for three-dimensional embedding; and we show that the new Lagrange multiplier spaces naturally lead to a class of easy-to-implement multigrid methods that achieve near optimal efficiency, as shown by numerical examples. For the specific case of a continuous Poisson coefficient in interface problems, we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints. As in Ill, this is used in a discontinuity removal technique that yields the standard 7-point stencil across the interface and only requires a modification to the right-hand side of the linear system. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

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