Solving partial differential equations by BKM combined with DDM

Abstract

The boundary knot method (BKM) has recently been developed as an inherently meshless, integration-free, boundary-type collocation technique for the numerical discretization of general partial differential equation systems. Unlike the method of fundamental solutions, the use of non-singular radial basis functions in the BKM avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain. But the condition number of the coefficient matrix of the linear algebraic equations which is derived from the homogeneous boundary value problem becomes very high as the total number of boundary knots increase, which in turn affects the accuracy of the computation result. Combined with domain decomposition method, we can avoid this problem easily. In this paper we will solve homogeneous and non-homogeneous partial differential equations using BKM combined with overlapped DDM. From the numerical experiments, we can find that it can solve PDEs accurately.