Feischl, M., Gantner, G., Haberl, A., Praetorius, D., & Führer, T. (2016). Adaptive boundary element methods for optimal convergence of point errors. Numerische Mathematik, 132(3), 541–567. https://doi.org/10.1007/s00211-015-0727-4
Applied Mathematics; Computational Mathematics; adaptive boundary element method; optimal convergence rates; point error; goal-oriented algorithm.
-
Abstract:
One particular strength of the boundary element method is that it allows for a high-order
pointwise approximation of the solution of the related partial differential equation via the
representation formula. However, the high-order convergence and hence accuracy
usually suffers from singularities of the Cauchy data. We propose two adaptive
mesh-refining algorithms and prove their quasi-optimal convergence behavior with
respect to the point error in the representation formula. Numerical examples for the
weakly-singular integral equations for the 2D and 3D Laplacian underline our theoretical
findings.