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A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow

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Abstract

Biharmonic equations have many applications, especially in fluid and solid mechanics, but is difficult to solve due to the fourth order derivatives in the differential equation. In this paper a fast second order accurate algorithm based on a finite difference discretization and a Cartesian grid is developed for two dimensional biharmonic equations on irregular domains with essential boundary conditions. The irregular domain is embedded into a rectangular region and the biharmonic equation is decoupled to two Poisson equations. An auxiliary unknown quantity Δu along the boundary is introduced so that fast Poisson solvers on irregular domains can be used. Non-trivial numerical examples show the efficiency of the proposed method. The number of iterations of the method is independent of the mesh size. Another key to the method is a new interpolation scheme to evaluate the residual of the Schur complement system. The new biharmonic solver has been applied to solve the incompressible Stokes flow on an irregular domain.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Guo Chen & Zhilin Li

  2. Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, 27695, USA

    Zhilin Li

  3. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Kent Ridge, Singapore, 117543

    Ping Lin

Authors
  1. Guo Chen
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  2. Zhilin Li
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  3. Ping Lin
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Correspondence to Zhilin Li.

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Communicated by Z. Chen.

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Chen, G., Li, Z. & Lin, P. A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow. Adv Comput Math 29, 113–133 (2008). https://doi.org/10.1007/s10444-007-9043-6

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  • DOI: https://doi.org/10.1007/s10444-007-9043-6

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