6. A Primer to Boundary Element Methods

This chapter introduces the BEM in its h −version. First we make Fourier expansion of Chap. 3 more precise by asymptotic error estimates. Then we prove direct and inverse approximation estimates for periodic spline approximation on curves. Hence we develop the analysis of Galerkin methods and collocation methods for Symm’s integral equation towards optimal a priori error estimates. Moreover, we subsume Galerkin and collocation methods as general projection methods. To this end we extend the above treatment of positive definite bilinear forms to the analysis of a sequence of linear operators that satisfy a uniform Gårding inequality and establish stability and optimal a priori error estimates in this more general setting. Interpreting several variants of collocation methods that combine collocation and quadrature as extended Galerkin methods we include their numerical analysis as well. Then augmenting the boundary element ansatz spaces by known singularity functions the Galerkin method is shown to converge with higher convergence rates. Finally to obtain higher convergence rates in weaker norms than the energy norm the Aubin–Nitsche duality estimates of FEM are extended to BEM so that it allows the incorporation of the singular solution expansion for nonsmooth domains. Sections 6.1–6.4 are based on the classroom notes by M. Costabel [116] whereas Sects. 6.5.1–6.5.6 are based on the classroom notes by W.L. Wendland [430]. Improved estimates of local type, pointwise estimates and postprocessing with the K-operator are considered in Sects. 6.5.7–6.5.9. Discrete collocation with trigonometric polynomials, where the concept of finite section operators is used, is a subject of Sect. 6.6. In Sect. 6.7 the standard BEM is enriched by special singularity functions modelling the behaviour of the solution near corners, thus yielding improved convergence. In Sect. 6.8 Galerkin-Petrov methods are considered. Section 6.9 presents the Arnold-Wendland approach to reformulate a collocation method as a Galerkin method whereas qualocation is investigated in Sect. 6.10. In Sect. 6.11 the use of radial basis functions (a meshless method) and of spherical splines in the Galerkin scheme is demonstrated for problems on the unit sphere. Integral equations of the first kind with the single layer and double layer potentials are our main subject. Integral equations of the second kind are studied only briefly, e.g. at the end of Sect. 6.4.