Tricomi's composition formula and the analysis of multiwavelet approximation methods for boundary integral equations

The present paper is mainly concerned with the convergence analysis of Galerkin-Petrov methods for the numerical solution of periodic pseudodifferential equations using wavelets and multiwavelets as trial functions and test functionals. Section 2 gives an overview on the symbol calculus of multidimensional singular integrals using Tricomi's composition formula. In Section 3 we formulate necessary and sufficient stability conditions in terms of the so-called numerical symbols and demonstrate applications to the Dirchlet problem for the Laplace equation

A discretization of Volterra integrals equations of the third kind with weakly singular kernels

In this paper we propose a method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations $$p(t)z(t) + int_0^t frac{h(t,tau)}{(t-tau)^{1-alpha}}z(tau) dtau = f(t),quad tin[0,1],enspace 0<alpha< 1.$$ Here $p(t)$ vanishes on some subset of $[t_1,t_2]subset[0,1]$ and mbox{$p(t) <delta$} for {mbox{$tin[t_1,t_2]$}}, where $delta$ is a sufficiently small positive number. The proposed method gives the accuracy $O(delta^{2nu/(2nu+1)})$ with respect to the mbox{$L_2$-norm,} where $nu$ is the parameter of sourcewise representation of the exact solution on $[t_1,t_2]$, and uses no more than $O(delta^{-(2-lambda)/alpha} log^{2+1/alpha}frac{1}{delta})$ values of Galer-kin functionals, where $lambdain(0,1/2)$ is determined in the act of choosing the regularization parameter within the framework of Morozov's discrepancy principle

Multiwavelet approximation methods for pseudodifferential equations on curves. Stability and convergence analysis

We develop a stability and convergence analysis of Galerkin-Petrov schemes based on a general setting of multiresolution generated by several refinable functions for the numerical solution of pseudodifferential equations on smooth closed curves. Particular realizations of such a multiresolution analysis are trial spaces generated by biorthogonal wavelets or by splines with multiple knots. The main result presents necessary and sufficient conditions for the stability of the numerical method in terms of the principal symbol of the pseudodifferential operator and the Fourier transforms of the generating multiscaling functions as well as of the test functionals. Moreover, optimal convergence rates for the approximate solutions in a range of Sobolev spaces are established

A discretization of Volterra integrals equations of the third kind with weakly singular kernels

In this paper we propose a method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations $$p(t)z(t) + int_0^t frac{h(t,tau)}{(t-tau)^{1-alpha}}z(tau) dtau = f(t),quad tin[0,1],enspace 0<alpha< 1.$$ Here $p(t)$ vanishes on some subset of $[t_1,t_2]subset[0,1]$ and mbox{$p(t) <delta$} for {mbox{$tin[t_1,t_2]$}}, where $delta$ is a sufficiently small positive number. The proposed method gives the accuracy $O(delta^{2nu/(2nu+1)})$ with respect to the mbox{$L_2$-norm,} where $nu$ is the parameter of sourcewise representation of the exact solution on $[t_1,t_2]$, and uses no more than $O(delta^{-(2-lambda)/alpha} log^{2+1/alpha}frac{1}{delta})$ values of Galer-kin functionals, where $lambdain(0,1/2)$ is determined in the act of choosing the regularization parameter within the framework of Morozov's discrepancy principle

A discretization of Volterra integrals equations of the third kind with weakly singular kernels

In this paper we propose a method of piecewise constant approximation for the solution of ill-posed third kind Volterra equations $$p(t)z(t) + int_0^t frac{h(t,tau)}{(t-tau)^{1-alpha}}z(tau) dtau = f(t),quad tin[0,1],enspace 0<alpha< 1.$$ Here $p(t)$ vanishes on some subset of $[t_1,t_2]subset[0,1]$ and mbox{$p(t) <delta$} for {mbox{$tin[t_1,t_2]$}}, where $delta$ is a sufficiently small positive number. The proposed method gives the accuracy $O(delta^{2nu/(2nu+1)})$ with respect to the mbox{$L_2$-norm,} where $nu$ is the parameter of sourcewise representation of the exact solution on $[t_1,t_2]$, and uses no more than $O(delta^{-(2-lambda)/alpha} log^{2+1/alpha}frac{1}{delta})$ values of Galer-kin functionals, where $lambdain(0,1/2)$ is determined in the act of choosing the regularization parameter within the framework of Morozov's discrepancy principle

On the characterization of self-regularization properties of a fully discrete projection method for Symm's integral equation

The influence of small perturbations in the kernel and the right-hand side of Symm's boundary integral equation, considered in an ill-posed setting, is analyzed. We propose a modification of a fully discrete projection method which is more economical in the sense of complexity and allows to obtain the optimal order of accuracy in the power scale with respect to the level of the noise in the kernel or in the parametric representation of the boundary

Fast computations with the harmonic Poincaré-Steklov operators on nested refined meshes

In this paper we develop asymptotically optimal algorithms for fast computations with the discrete harmonic Poincar'e-Steklov operators in presence of nested mesh refinement. For both interior and exterior problems the matrix-vector multiplication for the finite element approximations to the Poincar'e-Steklov operators is shown to have a complexity of the order O(Nreflog3N) where Nref is the number of degrees of freedom on the polygonal boundary under consideration and N = 2-p0 · Nref, p0 ≥ 1, is the dimension of a finest quasi-uniform level. The corresponding memory needs are estimated by O(Nreflog2N). The approach is based on the multilevel interface solver (as in the case of quasi-uniform meshes, see [20]) applied to the Schur complement reduction onto the nested refined interface associated with nonmatching decomposition of a polygon by rectangular substructures

Multilevel preconditioning on the refined interface and optimal boundary solvers for the Laplace equation

In this paper we propose and analyze some strategies to construct asymptotically optimal algorithms for solving boundary reductions of the Laplace equation in the interior and exterior of a polygon. The interior Dirichlet or Neumann problems are, in fact equivalent to a direct treatment of the Dirichlet-Neumann mapping or its inverse i.e. the Poincaré-Steklov (PS) operator. To construct a fast algorithm for the treatment of the discrete PS operator in the case of polygons composed of rectangles and regular right triangles, we apply the Bramble-Pasciak-Xu (BPX) multilevel preconditioner to the equivalent interface problem in the H1/2-setting. Furthermore, a fast matrix-vector multiplication algorithm is based on the frequency cutting techniques applied to the local Schur complements associated with the rectangular substructures specifying the nonmatching decomposition of a given polygon. The proposed compression scheme to compute the action of the discrete interior PS operator is shown to have a complexity of the order O(N logq N), q ∈ [2,3] with memory needs of O(N log2 N) where N is the number of degrees of freedom on the polygonal boundary under consideration. In the case of exterior problems we propose a modification of the standard direct BEM whose implementation is reduced to the wavelet approximation applied to either single layer or hypersingular harmonic potentials and, in addition, to the matrix-vector multiplication for the discrete interior PS operator

The qualocation method for Symm's integral equation on a polygon

This paper discusses the convergence of the qualocation method for Symm's integral equation on closed polygonal boundaries in ℝ2 . Qualocation is a Petrov-Galerkin method in which the outer integrals are performed numerically by special quadrature rules. Before discretisation a nonlinear parametrisation of the polygon is introduced which varies more slowly than arc-length near each corner and leads to a transformed integral equation with a regular solution. We prove that the qualocation method using smoothest splines of any order k on a uniform mesh (with respect to the new parameter) converges with optimal order O(hk ). Furthermore, the method is shown to produce superconvergent approximations to linear functionals, retaining the same high convergence rates as in the case of a smooth curve

A multiscale method for the double layer potential equation on a polyhedron

This paper is concerned with the numerical solution of the double layer potential equation on polyhedra. Specifically, we consider collocation schemes based on multiscale decompositions of piecewise linear finite element spaces defined on polyhedra. An essential difficulty is that the resulting linear systems are not sparse. However, for uniform grids and periodic problems one can show that the use of multiscale bases gives rise to matrices that can be well approximated by sparse matrices in such a way that the solutions to the perturbed equations exhibits still sufficient accuracy. Our objective is to explore to what extent the presence of corners and edges in the domain as well as the lack of uniform discretizations affects the performance of such schemes. Here we propose a concrete algorithm, describe its ingredients, discuss some consequences, future perspectives, and open questions, and present the results of numerical experiments for several test domains including non-convex domains