Multiscale methods for the solution of the Helmholtz and Laplace equations

This paper presents some numerical results about applications of multiscale techniques to boundary integral equations. The numerical schemes developed here are to some extent based on the results of the papers [6]—[10]. Section 2 deals with a short description of the theory of generalized Petrov-Galerkin methods for elliptic periodic pseudodifferential equations in $\mathbb{R}^n$ covering classical Galerkin schemes, collocation, and other methods. A general setting of multiresolution analysis generated by periodized scaling functions as well as a general stability and convergence theory for such a framework is outlined. The key to the stability analysis is a local principle due to one of the authors. Its applicability relies here on a sufficiently general version of a so-called discrete commutator property of wavelet bases (see [6]). These results establish important prerequisites for developing and analysing methods for the fast solution of the resulting linear systems (Section 2.4). The crucial fact which is exploited by these methods is that the stiffness matrices relative to an appropriate wavelet basis can be approximated well by a sparse matrix while the solution to the perturbed problem still exhibits the same asymptotic accuracy as the solution to the full discrete problem. It can be shown (see [7]) that the amount of the overall computational work which is needed to realize a required accuracy is of the order $\mathcal{O}(N(\log N)^b)$, where $N$ is the number of unknowns and $b \geq 0$ is some real number

Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM

Elliptic boundary value problems which are posed on a random domain can be mapped to a fixed, nominal domain. The randomness is thus transferred to the diffusion matrix and the loading. While this domain mapping method is quite efficient for theory and practice, since only a single domain discretisation is needed, it also requires the knowledge of the domain mapping. However, in certain applications, the random domain is only described by its random boundary, while the quantity of interest is defined on a fixed, deterministic subdomain. In this setting, it thus becomes necessary to compute a random domain mapping on the whole domain, such that the domain mapping is the identity on the fixed subdomain and maps the boundary of the chosen fixed, nominal domain on to the random boundary. To overcome the necessity of computing such a mapping, we therefore couple the finite element method on the fixed subdomain with the boundary element method on the random boundary. We verify the required regularity of the solution with respect to the random domain mapping for the use of multilevel quadrature, derive the coupling formulation, and show by numerical results that the approach is feasible

Fast wavelet collocation methods for second kind integral equations on polygons

In this thesis we develop fast wavelet collocation methods for integral equations of the second kind with weakly singular kernels over polygons . For this purpose, we construct multiscale wavelet functions and collocation functionals having vanishing moments. Moreover, we propose several truncation strategies, which lead to fast algorithms, for the coefficient matrix of the corresponding discrete system. Critical issues for numerical implementation of such methods are considered, such as choices of practical truncation strategies, numerical integration of weakly singular integrals, error controls of numerical quadrature and numerical solutions of resulting compressed linear systems. Numerical experiments are given to demonstrate proposed ideas and methods. Finally, parallel computing using developed methods is investigated.;That this work received partial support from the US NSF grant EPSCoR-0132740

Quadrature methods for 2D and 3D problems

AbstractIn this paper we give an overview on well-known stability and convergence results for simple quadrature methods based on low-order composite quadrature rules and applied to the numerical solution of integral equations over smooth manifolds. First, we explain the methods for the case of second-kind equations. Then we discuss what is known for the analysis of pseudodifferential equations. We explain why these simple methods are not recommended for integral equations over domains with dimension higher than one. Finally, for the solution of a two-dimensional singular integral equation, we prove a new result on a quadrature method based on product rules