Solving Eigenvalue Problems of the Biharmonic Operator on Manifolds with Boundaries
Date Issued
2012
Date
2012
Author(s)
Chern, Ren-Haur
Abstract
The numerical eigenvalue problem for the biharmonic operator with clamped boundary condition on a curved manifold is considered in this article. The recently developed closest point method is adopted. The closest point method for solving PDEs on a manifold is to solve an equivalent problem in a neighborhood of the manifold in the Euclidean space the manifold embedded in, where finite difference method can be easily applied. In the present study, the main difficulty lies on the clamped boundary condition. A sophisticated but natural extrapolation method is introduced for the clamped boundary condition, which is a second order method theoretically; however, the present numerical tests show that the accuracy is less than second order. Our numerical investigation shows the coefficient of the truncation error from the boundary is too large due to the high order extrapolation, yielding the discrepancy between the theoretical and numerical results.
Subjects
Closest point method
Eigenvalue problems
Biharmonic operator
Clamped boundary condition
PDEs on manifolds
Type
thesis
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