Implementation and study of boundary integral operators related to PDE:s in the plane

Abstract: The method of solving boundary value problems of partial differential equations numerically by first reformulating the problem as a boundary integral equation has many advantages over other methods, but also some unique difficulties. Some of these difficulties stem from problems in evaluating singular or nearly singular integral operators, and solving these difficulties is an active field of research. Known results are summarized, and an accessible program package is developed, using underlying Gauss--Legendre quadrature and product integration, which can be applied to boundary value problems with smooth boundaries. The program package is available on GitHub at the link https://github.com/erikandersson98/BIE-CELib. Different methods of implementing integral operators related to the Laplace and Helmholtz equations are compared with regards to accuracy and convergence rate, both when used in boundary value problems, and when applied to theoretical identities. The methods are based on product integration, as well as on global and local regularization. To conclude, recommended implementations based on the results are given, as well as possible directions to expand the package.