Dirac delta methods for Helmholtz transmission problems

Abstract

In this paper we use a boundary integral method with single layer potentials to solve a class of Helmholtz transmission problems in the plane. We propose and analyze a novel and very simple quadrature method to solve numerically the equivalent system of integral equations which provides an approximation of the solution of the original problem with linear convergence (quadratic in some special cases). Furthermore, we also investigate a modified quadrature approximation based on the ideas of qualocation methods. This new scheme is again extremely simple to implement and has order three in weak norms. Mathematics Subject Classifications (2000) 65R20 · 65N38

Boundary Element Simulation of Thermal Waves

“…The choice ε = 0 is not valid, since the kernel functions V λ p p and V λ + pp are singular at s = t. The choices ε = ±1/2 give the same method, which is not stable: testing on the midpoint between two charges gives a method that does not work properly. This fact is derived from a factorization of the discrete equations like in (36), from where we see that we have to separately consider stability of two methods, one for integral equations with logarithmic singularities in the kernel and another one for integral equations of the second kind. In [22,94] it is proven that the choices ε = ±1/2 are unstable and that ε = ±1/6 are superconvergent in a sense we will explain later on.…”

“…Therefore, convergence can be studied for the principal part of the operator (34), which is a reordering of P 2 × 2 blocks, each one taking place on an interface p (35). The second decomposition in (36) can be also applied at the discrete level.…”

Convergence analysis of a high-order Nyström integral-equation method for surface scattering problems

“…The remainder is the singular part of the kernel, 14) which, like the kernel K 1 , can be integrated accurately by means of a polar change of variables; see Remark 2.1. The parameter δ, which controls the support of the kernel K sing,δ (r, · ), plays an essential role in both, the performance of the algorithm and its theoretical analysis; see Remark 2.4 for details.…”

“…To do this, we introduce the discrete operators 17) and, for ξ ∈ C 0 (I 2 ), we define (cf. [10] and [14])…”

“…In particular, for the sake of the analysis we introduce Fourier projection operators, bi-periodic trigonometric interpolation operators, and a discrete operator [10] that produces a linear combination of Dirac delta distributions on a uniform grid from a smooth function input. The convergence analysis for the operators arising from the regular part of the original boundary integral operator follows the lines of the theory [10,14] on periodic integral equations in one variable. The final (rather technical) element of our convergence proof is a detailed analysis of the integration error arising from the numerical polar coordinate integration algorithm for products of smooth functions, "shrinking" floating cut-off functions and bi-variate trigonometric polynomials, in terms of Sobolev norms of the latter polynomials.…”