Boundary element approach for an inverse source problem of the Poisson equation with a one-point-mass-like source
The authors consider the problem of determining the source term \(f\) of a two-dimensional Poisson equation \(-\Delta u = f\) in a bounded convex domain, assuming the solution \(u\) and the flux given on the boundary. They discuss in detail the particular problem of finding the center \((a,b)\) of a radially symmetric source \(f(x,y) = F(r_{a,b})\), \(r^ 2_{a, b} = (x - a)^ 2 + (y - b)^ 2\), with \(a\) and \(b\) unknown. They eliminate the harmonic part of \(u\) by using boundary integral relations and propose an iterative algorithm for approximating the center. For approximation of the logarithmic potential \(u\) on the boundary they use the boundary element method and derive error estimates, under appropriate smoothness assumptions, for the case of piecewise constant ansatz. Finally, they demonstrate applicability of their method by displaying results of a numerical case study.
scientific article; zbMATH DE number 622120
scientific article