A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow |
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Authors: | Guo Chen Zhilin Li Ping Lin |
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Institution: | (1) Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA;(2) Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695, USA;(3) Department of Mathematics, National University of Singapore, 2 Science Drive 2, Kent Ridge, Singapore, 117543 |
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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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Keywords: | Biharmonic equation Irregular domain Augmented method Immersed interface method Incompressible Stokes flow |
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