Well-posedness and Wavelet-based Approximations for Hypersingular Integral Equations

Abstract

<p>The main object of this thesis is to investigate the hypersingular integral equations which arise in Boundary Element (BE) models for crack problems. Although the associated hypersingular integrals are not defined in the usual sense, we interpret them in terms of Hadamard finite part integral operators or as pseudo-differential operators in the distribution sense.</p> <p>By introducing weighted Sobolev spaces to regularize the equations, we have proved the well-posedness for such hypersingular integral equations. Global error estimates are obtained in the thesis.</p> <p>A new approach to the numerical solution of the hypersingular equations based on the recently developed theory of wavelets is presented. Rather than applying wavelet bases directly to obtain new discretizations, we exploit the wavelet bases to obtain more efficient solution algorithms for the more classical discretizations. We discretize the hypersingular integral equation using the piecewise polynomial collocation method. The discrete wavelet transform is then used for the resulting dense algebraic system. This procedure involves O(N²) operations and leads to a sparse matrix problem. Solving this sparse matrix system requires only O(N log² N) operations whereas O(N³) operations are required for the traditional. BE method. Exploiting an indexed storage structure, we reduce the memory requirements from O(N²) to O(N log N) words. Furthermore, the method is applicable to all operator or matrix (with arbitrary geometries) problems as long as the operator or matrix possesses only a finite number of singularities in some rows or columns.</p> <p>We demonstrate that the wavelet-based method can be extended to higher dimensional integral equation problems because these equations can also be discretized by the piecewise polynomial collocation approximation.</p>