Conforming boundary element methods in acoustics

Conforming boundary element methods are developed for three-dimensional acoustic wave scattering by sound hard (Neumann problem) and sound soft (Dirichlet problem) objects. Theoretically, these methods guarantee convergence of the solution in the norm of an appropriate function space as the number of degrees of freedom is increased. For the integral equations of the first kind conforming boundary elements can be obtained with conventional Galerkin׳s method and finite element spaces (FE). However, for the integral equations of the second kind this standard technique does not lead to a conforming method, rather Petrov–Galerkin׳s methods, where the basis and testing functions are not identical, are required. In the numerical implementation of the Petrov–Galerkin methods special “dual” FE spaces defined on the barycentrically refined mesh are used to test the equations. These dual FE spaces also play an important role in the Calderon matrix multiplicative preconditioners applied to regularize ill-conditioned integral equations of the first kind.