Immersed Interface Method for Biharmonic Equations on Irregular Domain and Its Applications
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Date
2004-12-30
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Abstract
This thesis presents a fast algorithm for solving two-dimensional biharmonic equations on irregular domains. To avoid mesh generation difficulties associated with unstructured, body fitted grid, the irregular domain is embedded into a uniform Cartesian grid. The biharmonic equation is decomposed into two coupled Poisson equations.
The solution of the coupled Poisson system depends on the Laplacian Δu on the boundary. We use a weighted least squares interpolation to approximate the Laplacian on the boundary from inside of the region. The accuracy of the interpolation scheme turns out to be a crucial step in solving the biharmonic problem for our algorithm.
The resulting linear system involves both the solution and the Laplacian on the boundary. In order to take advantage of fast Poisson solvers, we use Generalized Minimum Residue method to solve for Δu, and use the Immersed Interface Method to solve the coupled Poisson problem. Putting all these techniques together, we get a second order fast algorithm for solving biharmonic equations on irregular domains. Numerical analysis show the algorithm is very stable and the number of iterations of our method seems to be independent of the mesh size.
We also investigate some applications of the proposed fast algorithm for incompressible Stokes equations and the biharmonic equation with a linear parameter.
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Keywords
immersed interface method, biharmonic equation, GMRES
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Degree
PhD
Discipline
Applied Mathematics
