Abstract
In this study, a high accuracy numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations on a spherical domain is developed. By employing the orthogonal spherical polynomials approximation and the spectral theory of compact operator, the error estimates of approximate eigenvalues and eigenfunctions are provided. By adopting orthogonal spherical base functions, the discrete model with sparse mass and stiff matrices is established so that it is very efficient for finding the numerical solutions of biharmonic eigenvalue equations on the spherical domain. Some numerical examples are provided to validate the theoretical results.
Highlights
In this article, we consider the following biharmonic eigenvalue equations: u = λu, in, ( )∂u u = =, on ∂, ∂n where ⊂ Rd (d =, )is an open disk or a ball,∂u ∂n denotes the outer normal derivative of u on ∂
To the best of our knowledge, there is not any article on a high accuracy numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations on the spherical domain
The task of this paper is to develop a high precision numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations in the spherical domain
Summary
We consider the following biharmonic eigenvalue equations:. ∂u ∂n denotes the outer normal derivative of u on ∂. All methods mentioned above are low-order finite element methods so that it is very difficultly and expensively to obtain high accuracy numerical solutions, especially for the three-dimensional spherical domain. To the best of our knowledge, there is not any article on a high accuracy numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations on the spherical domain. The task of this paper is to develop a high precision numerical method based on the spectral theory of compact operator for biharmonic eigenvalue equations in the spherical domain. In Section , by adopting orthogonal spherical base functions, we establish the discrete model with sparse mass and stiff matrices which is very efficient for finding the numerical solutions of biharmonic eigenvalue equations on the spherical domain.
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