Remarks on the Gaussian quadrature rule for finite-part integrals with a second-order singularity

DOI10.1016/0045-7825(88)90045-XzbMath0646.65018OpenAlexW2337093020MaRDI QIDQ1104036

B. George

Publication date: 1988

Published in: Computer Methods in Applied Mechanics and Engineering (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0045-7825(88)90045-x

zbMATH Keywords

convergencenumerical resultshypersingular integral equationsfinite-part integralsGaussian quadrature ruleGauss-Legendre quadrature rulesecond-order singularityis paper deals with preconditioning techniques for conjugate gradient methods applied to the solution of a linear system \(Ax=b\) where A is a symmetric positive definite sparse matrix coming from a finite element or a finite difference approximation method of a partial differential equation on a rectangular domain with Dirichlet boundary condition. The preconditioning techniques are studied here are all based on an incomplete Cholesky decomposition. One method differs from another one by the way of grid point ordering and more generally of the degrees of freedom or unknowns. The authors study the classical ordering methods, or coloring methods. Then they propose a systematic coloring strategy permitting to reduce the essential part of the computation at each preconditioned conjugate gradient iteration to operations in ``long length vectors. Thus the degree of vectorization can be very satisfactory. The authors apply their coloring strategy to four test examples. They compare their results with some results taken from the literature

Mathematics Subject Classification ID

Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane (30E20) Numerical methods for integral equations (65R20) Approximate quadratures (41A55) Numerical quadrature and cubature formulas (65D32) Brittle damage (74R05) Integral equations with kernels of Cauchy type (45E05)

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