Commun. Comput. Phys., 12 (2012), pp. 515-527.

An Adaptive Mesh Refinement Strategy for Immersed Boundary/Interface Methods

Zhilin Li 1*, Peng Song 2

1 Center for Research in Scientific Computation and Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA; and Nanjing Normal University, China.
2 Operations Research Program, North Carolina State University, Raleigh, NC 27695, USA.

Received 7 February 2011; Accepted (in revised version) 15 August 2011
Available online 20 February 2012


An adaptive mesh refinement strategy is proposed in this paper for the Immersed Boundary and Immersed Interface methods for two-dimensional elliptic interface problems involving singular sources. The interface is represented by the zero level set of a Lipschitz function $\varphi(x,y)$. Our adaptive mesh refinement is done within a small tube of $|\varphi(x,y)|\le \delta$ with finer Cartesian meshes. The discrete linear system of equations is solved by a multigrid solver. The AMR methods could obtain solutions with accuracy that is similar to those on a uniform fine grid by distributing the mesh more economically, therefore, reduce the size of the linear system of the equations. Numerical examples presented show the efficiency of the grid refinement strategy.

AMS subject classifications: 52B10, 65D18, 68U05, 68U07

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Key words: Adaptive mesh refinement, immersed boundary method, immersed interface method, elliptic interface problem, Cartesian grid method, level set representation, singular sources.

*Corresponding author.
Email: (Z. Li), (P. Song)

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