Hybrid Galerkin boundary elements: theory and implementation

Tn this paper we present a new quadrature method for computing Galerkin sti ffness matrices arising from the discretisation of 3D boundary integral equ ations using continuous piecewise linear boundary elements. This rule takes as points some subset of the nodes of the mesh and can be used for computi ng non-singular Galerkin integrals cor-responding to pairs of basis functio ns with non-intersecting supports. When this new rule is combined with stan dard methods for the singular Galerkin integrals we obtain a "hybrid" Galer kin method which has the same stability and asymptotic convergence properti es as the true Galerkin method but a complexity more akin to that of a coll ocation or Nystrom method. The method can be applied to a wide range of sin gular and weakly-singular first- and second-kind equations, including many for which the classical Nystrom method is not even defined. The results app ly to equations on piecewise-smooth Lipschitz boundaries, and to non-quasiu niform (but shape-regular) meshes. A by-product of the analysis is a stabil ity theory for quadrature rules of precision 1 and 2 based on arbitrary poi nts in the plane. Numerical experiments demonstrate that the new method rea lises the performance expected from the theory. Mathematics Subject Classif ication ((1991): 65N38.