Search

There has been considerable attention given in recent years to the problem of extending finite and boundary element-based analysis of Helmholtz problems to higher frequencies. One approach is the Partition of Unity Method, which has been applied successfully to boundary integral solutions of Helmholtz problems, providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy. These benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions. In this paper we adapt the numerical steepest descent method to evaluate these integrals for two-dimensional problems. The approach is successful in reducing the computational effort for most integrals encountered. The paper includes some numerical features that are important for successful practical implementation of the algorithm.

The domain decomposition method proposed by Schwarz [1] is gaining significance in view of the possibilities for parallel implementation. In this paper, we apply the domain decomposition method to the biharmonic equation in two overlapping discs. In each subdomain, a boundary integral formula is used to obtain the solution. Unlike the finite element or finite difference methods where the iterations have to be performed on the entire domain, the iterations of the boundary integral formulation need to be done only at the discretized points of the boundaries. Another aspect of the paper is that, in order to handle the operations with dense matrices arising from the boundary integral formulation, the discrete wavelet transform is used to compress the matrices which results in reduced computations without loss of accuracy. Examples involving a Laplace equation in an L-shaped region and a domain bounded by a cardioid and a circle are also illustrated.

In this paper we analyze a family of full discretizations of spline Galerkin methods for a class of systems of boundary integral equations of the first kind with logarithmic principal part. We prove the existence of an asymptotic expansion of the error of the Galerkin and the optimal order Galerkin collocation method. We finally derive asymptotic expansions for some common postprocessings of the solutions, both exactly and under the effect of additional discretization. Some examples where these techniques apply are provided.

Standard implementations of the fast multipole method, which compute fields due to point sources or dipoles, cannot be used to accurately evaluate the single- and double-layer potentials of potential theory close to the boundary, or on the boundary when the boundary curves back on itself. We describe the modifications necessary to accurately evaluate layer potentials in two dimensions, which include quadrature rules for the short-range contributions to the field, continuous multipole moments for long-range contributions, and a more complex bookkeeping procedure. We give formulae for second-, third-, and fourth-order methods. We show tests to verify the correctness of the method and numerical results which demonstrate the usefulness of the method for evaluating layer potentials near the boundary.