A meshless solution for potential equations using a continuous-valued circular line source

Abstract

We find solutions to Δ2U = 0 in a simply-connected 2-D domain D, using a continuous line source associated with a concentration function comprising n undetermined parameters. This choice reduces ill-conditioning effects by reducing the number of parameters involved. The choice of a continuous circular line source C around D follows from previous results indicating that, when solving the same problem with discrete point sources, the result is independent of precise placement of sources. The circle is associated with a concentration function that is constrained to satisfy the problem's boundary conditions. Accuracy is achievable using a number of parameters which, had discrete sources been used, would be insufficient to represent the geometry of D, thus giving inaccurate results. Empirical investigations with various forms of concentration function show that with some domains, the error in calculated values of U can be less than 0.1%: an order of magnitude improvement over discrete methods. More complex domains yield less accuracy, and, after testing on a range of domains, we formulate an empirical rule for an appropriate form for the concentration function for a generic domain. Code requiring high-precision arithmetic was developed in Mathematica, which also simplifies routine tasks of solving linear systems and integrations.