A Wavelet Algorithm for the Solution of a Singular Integral Equation over a Smooth Two-dimensional Manifold

In this paper we consider a piecewise bilinear collocation method for the solution of a singular integral equation over a smooth surface. Using a fixed set of parametrizations, we introduce special wavelet bases for the spaces of test and trial functions. The trial wavelets have two vanishing moments only if their supports do not intersect the lines belonging to the common boundary of two subsurfaces defined by different parameter representations. Nevertheless, analogously to well-known results on wavelet algorithms, the stiffness matrices with respect to these bases can be compressed to sparse matrices such that the iterative solution of the matrix equations becomes fast. Finally, we present a fast quadrature algorithm for the computation of the compressed stiffness matrix

Error bounds of certain Gaussian quadrature formulae

We study the kernel of the remainder term of Gauss quadrature rules for analytic functions with respect to one class of Bernstein-Szego weight functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds of the corresponding Gauss quadratures

Error bounds of the Micchelli-Sharma quadrature formula for analytic functions

Micchelli and Sharma constructed in their paper [On a problem of Turan: multiple node Gaussian quadrature, Rend. Mat. 3 (1983) 529-552] a quadrature formula for the Fourier-Chebyshev coefficients, which has the highest possible precision. For analytic functions the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points not equal 1 and a sum of semi-axes rho > 1, for the quoted quadrature formula. Starting from the explicit expression of the kernel, we determine the location on the ellipses where the maximum modulus of the kernel is attained, and derive effective error bounds for this quadrature formula. Numerical examples are included

Error bounds of the Micchelli-Sharma quadrature formula for analytic functions

Micchelli and Sharma constructed in their paper [On a problem of Turan: multiple node Gaussian quadrature, Rend. Mat. 3 (1983) 529-552] a quadrature formula for the Fourier-Chebyshev coefficients, which has the highest possible precision. For analytic functions the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points not equal 1 and a sum of semi-axes rho > 1, for the quoted quadrature formula. Starting from the explicit expression of the kernel, we determine the location on the ellipses where the maximum modulus of the kernel is attained, and derive effective error bounds for this quadrature formula. Numerical examples are included

Error bounds of certain Gaussian quadrature formulae

AbstractWe study the kernel of the remainder term of Gauss quadrature rules for analytic functions with respect to one class of Bernstein–Szegö weight functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective error bounds of the corresponding Gauss quadratures

The error bounds of gauss-lobatto quadratures for weights ofbernstein-szego type

In this paper, we consider the Gauss-Lobatto quadrature formulas for the Bernstein-Szego weights, i.e., any of the four Chebyshev weights divided by a polynomial of the form rho(t) = 1 - 4 gamma/(1+gamma)(2) t(2), where t is an element of (-1,1) and gamma is an element of (-1,0]. Our objective is to study the kernel in the contour integral representation of the remainder term and to locate the points on elliptic contours where the modulus of the kernel is maximal. We use this to derive the error bounds for mentioned quadrature formulas

The error bounds of gauss-lobatto quadratures for weights ofbernstein-szego type

In this paper, we consider the Gauss-Lobatto quadrature formulas for the Bernstein-Szego weights, i.e., any of the four Chebyshev weights divided by a polynomial of the form rho(t) = 1 - 4 gamma/(1+gamma)(2) t(2), where t is an element of (-1,1) and gamma is an element of (-1,0]. Our objective is to study the kernel in the contour integral representation of the remainder term and to locate the points on elliptic contours where the modulus of the kernel is maximal. We use this to derive the error bounds for mentioned quadrature formulas

Error estimates of anti-Gaussian quadrature formulae

Anti-Gauss quadrature formulae associated with four classical Chebyshev weight functions are considered. Complex-variable methods are used to obtain expansions of the error in anti-Gaussian quadrature formulae over the interval vertical bar-1, 1 vertical bar. The kernel of the remainder term in anti-Gaussian quadrature formulae is analyzed. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L-infinity-error bounds of anti-Gauss quadratures. Moreover, the effective L-1-error estimates are also derived. The results obtained here are an analogue of some results of Gautschi and Varga (1983) [11], Gautschi et al. (1990) [9] and Hunter (1995) [10] concerning Gaussian quadratures

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