Singularity problems from source functions as source nodes located near boundaries; numerical methods and removal techniques

Abstract

Consider the Dirichlet problem for Laplace’s/Poisson’s equation in a bounded simply-connected domain S. The source function qln|PQ*¯| is a fundamental solution (FS), and it can be found in many physical problems. The singularity occurs when the boundary value data affected by qln|PQ*¯| as the source node Q* is located near the boundary Γ(=∂S). So far, there is no comprehensive study on this kind of singularity. In this paper, the solution singularity is explored and the reduced convergence rates are derived for the method of particular solutions (MPS) and the method of fundamental solutions (MFS). Classic domains, such as disks, ellipses and polygons, are discussed for analysis and computation. For this new kind of solution singularity, the convergence rates of the MFS and the MPS are very low. The errors caused by numerical integration are critical to the solution accuracy. A new analytic framework for the collocation Trefftz method (CTM) involving numerical integration is established in this paper; this is an advanced development of our previous study [19]. Since the numerical solutions are poor in accuracy, removal techniques are essential in applications. New removal techniques are proposed for a node Q* located near Γ. In this paper, an additional FS as, d0ln|PQ0¯|, is added to the original source nodes in the traditional MFS, and the point charge d0(=q) and the source node Q0 are unknowns to be sought by nonlinear solvers (such as the secant method). When the source node Q* is located inside S but near Γ, both simple domains (such as disks, ellipses and squares) and complicated domains (such as amoeba-like domains) are studied. The validity of the new removal techniques is supported by numerical experiments. The removal techniques in this paper may also be applied to solve source identification problems. A comprehensive study has been completed in this paper for the solitary source function qln|PQ*¯| as the source node Q* is located near Γ.