New locations of source nodes for method of fundamental solutions solving Laplace’s equation; pseudo radial-lines

Abstract

Consider 2D Laplace’s equation in a bounded simply-connected domain S, and solve it by the method of fundamental solutions (MFS). The source nodes must be located outside the domain boundary Γ(=∂S). How to select better source nodes is essential to the MFS in both theory and computation. In this paper, we study new locations of source nodes along pseudo radial-lines outside Γ. Denote the source nodes by (Ri,ϕi∗) in polar coordinates, where Ri>ρmax=maxρ|Γ but angle ϕi∗ is fixed. We propose new algorithms using two pseudo radial-lines. Two different angles ϕ1∗ and ϕ2∗ are fixed, and the source nodes as (Ri,ϕ1∗) and (Ri,ϕ2∗) are chosen, where Ri∈[R0,Rmax]. Adaptive error estimates are derived by using the numerical coefficients obtained. We study two kinds of locations of source nodes, Case I: uniform Ri∈[R0,Rmax] and Case II: uniform xi=ρmaxRi∈[a,b], where a=ρmaxRmax and b=ρmaxR0. Locations of source nodes are defined by four parameters (i.e., R0,Rmax,ϕ1∗ and ϕ2∗) in Cases I and II. Numerical experiments are carried out for Cases I and II, and better parameters are found. Two radial-lines of Case II are antenna-like and radar-line-like. Based on the sensitivity index, Case II offers better numerical performance. The algorithms of two radial-lines converge fast; both Cases I and II are satisfactory for scientific/engineering problems. In summary, the algorithms for new two pseudo radial-lines in Cases I and II have been established to enrich the MFS family, and the new algorithms and techniques seeking better source nodes in this paper are further developments of the MFS.