The Local RBF-FD Method For Laplace Eigenvalue Problem

Differential operator eigenvalue problem is an important problem in partial differential equations.Its research involves many important aspects,such as spectral analysis and numerical methods.It is closely related to many other disciplines in mathematics,and is widely used in physics,chemistry,biology.As the most common type of differential operator eigenvalue problem,the study of Laplace eigenvalue problem is very important.It has rich theoretical results and practical physics background in mathematics and physical sciences.This paper mainly considers the numerical method of Laplace eigenvalue problem,and chooses a new type of meshless method,that is the radial basis functiongenerated finite difference method.This method is isotropic,simple in form,meshless and Irrelevant in spatial dimension.The main work of the paper: the radial basis function-generated finite difference scheme of the eigenvalue problem is studied,which makes the Laplace eigenvalue problem into a discrete matrix eigenvalue problem.Then the appropriate numerical method is used to study the eigenvalues and eigenvectors of the matrix.In the numerical examples,we select the rectangular region and the L-shape region as the solution domain of the eigenvalue problem,and perform calculation in the case of three different node layouts.Comparing the different radial basis functions in particular and studying the influence of the shape parameter on the error precision.Through numerical examples,the effectiveness of the algorithm in solving the Laplace eigenvalue problem is verified.