Condition numbers and local errors in the boundary element method

In this chapter we investigate local errors and condition numbers in the BEM. The results of these investigations are important in guiding adaptive meshing strategies and solvability of linear systems in BEM. We show that the local error for the BEM with constant or linear elements decreases quadratically with the boundary element mesh size. We also investigate better ways of treating boundary conditions to reduce the local errors. The results of our numerical experiments confirm the theory. The values of the condition numbers of the matrices that appear in the BEM depend on the shape and size of the domain on which a problem is defined. For certain critical domains these condition numbers can even become infinitely large. We show that this holds for several classes of boundary value problems and propose a number of strategies to guarantee moderate condition numbers.