Wavelet Boundary Element Methods: Adaptivity and Goal-Oriented Error Estimation

This article is dedicated to the adaptive wavelet boundary element method. It computes an approximation to the unknown solution of the boundary integral equation under consideration with a rate N_{dof}^{−s}, whenever the solution can be approximated with this rate in the setting determined by the underlying wavelet basis. The computational cost scale linearly in the number N_{dof} of degrees of freedom. Goal-oriented error estimation for evaluating linear output functionals of the solution is also considered. An algorithm is proposed that approximately evaluates a linear output functional with a rate N_{dof}^{−(s+t)}, whenever the primal solution can be approximated with a rate N_{dof} and the dual solution can be approximated with a rate N_{dof}^{−t}, while the cost still scale linearly in N_{dof}. Numerical results for an acoustic scattering problem and for the point evaluation of the potential in case of the Laplace equation are reported to validate and quantify the approach.