Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations

Abstract

This paper makes a first attempt to use a new localized method of fundamental solutions (LMFS) to accurately and stably solve the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations in complex geometries. The LMFS firstly divides the whole physical domain into several small overlapping subdomains, and then employs the traditional method of fundamental solutions (MFS) formulation in every local subdomain for calculating the unknown coefficients on the local fictitious boundary. After that, a sparse linear system is formed by using the governing equation for interior nodes and the nodes on under-specified boundary, and by using the given boundary conditions for the nodes on over-specified boundary. Finally, the numerical solutions of the inverse problems can be obtained by solving the resultant sparse system. Compared with the traditional MFS with the “global” boundary discretization, the LMFS requires less computational cost, which may make the LMFS suitable for solving large-scale problems. Numerical experiments demonstrate the validity and accuracy of the proposed LMFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations with noisy boundary data.