Advanced methods of boundary integral equations for Laplace’s equation: Relations to method of fundamental solutions

Consider Laplace’s equation in a bounded simply-connected domain $S$. The harmonic functions in $S$ can be represented by boundary integral equations of the first kind (BIE), which is defined on the domain boundary $\Gamma (=\partial S)$. The unknowns are solved via the source nodes $Q$ on $\Gamma$, and their discrete algorithms are called the conventional boundary integral equation method (BIEM). From the Sobolev extension theory, the harmonic solutions can be extended to a larger simply-connected domain $S_R$ with $S\subset S_R$. We renovate the BIEM by moving the source nodes $Q$ from $\Gamma$ to ${\Gamma }_R$, to eliminate the singularity of the BIEM. Their discrete algorithms are simple and straightforward, and called the discrete boundary integral equation method (DBIEM). The main effort of this paper is to explore the error analysis of the DBIEM via Galerkin approaches. An important application of the error analysis is for the method of fundamental solutions (MFS), where the source nodes $Q$ are also located on ${\Gamma }_R$. The error analysis of the MFS in Dou et al. (2020) , Li (2009), Li et al. (2023) is confined to circular/elliptic pseudo-boundaries. The error analysis of the MFS for non-circular/non-elliptic pseudo-boundaries can be obtained from this paper. Since the pseudo-boundary ${\Gamma }_R$ is different from the boundary $\Gamma$ in the conventional BIEM, the inf-sup lemma is different from that in the boundary element method (BEM). The first work is to prove the new inf-sup lemma. For polygonal domains, when polygonal pseudo-boundaries are chosen, the piecewise interpolation polynomials of low orders are used as in the finite element method (FEM) and the BEM. The error bounds are derived for smooth solutions to achieve the optimal convergence rates. This error analysis is also valid for smooth $\Gamma$, and the constant/linear and linear/quadratic elements may find wide applications. The error analysis can also be applied to the MFS. Numerical experiments are reported to verify the analysis made. This paper is essential to the error analysis of both the DBIEM and the MFS.