Fast and accurate solvers for weakly singular integral equations

• Published in: Numerical algorithms Vol. 92; no. 4; pp. 2045 - 2070
• Main Authors: Grammont, Laurence, Kulkarni, Rekha P., Vasconcelos, Paulo B.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2023; Springer Nature B.V; Springer Verlag
• Subjects: Algebra; Algorithms; Approximation; Collocation methods; Computer Science; Integral equations; Iterative methods; Mathematical analysis; Mathematics; Methods; Numeric Computing; Numerical Analysis; Original Paper; Polynomials; Singular integral equations; Singularity (mathematics); Theory of Computation; Product integration; 65R20; Algebraic singularity; Logarithmic singularity; Graded mesh; 45L05; Collocation method; Interpolatory projection; 65J15; Piecewise polynomial approximation; Weakly singular integral operator
• ISSN: 1017-1398; 1572-9265; 1572-9265
• DOI: 10.1007/s11075-022-01376-x

Consider an integral equation λ u - T u = f , where T is an integral operator, defined on C [0, 1],  with a kernel having an algebraic or a logarithmic singularity. Let π m denote an interpolatory projection onto a space of piecewise polynomials of degree ≤ r - 1 with respect to a graded partition of [0, 1] consisting of m subintervals. In the product integration method, an approximate solution is obtained by solving λ u m - T π m u m = f . As in order to achieve a desired accuracy, one may have to choose m large, we find approximations of u m using a discrete modified projection method and its iterative version. We define a two-grid iteration scheme based on this method and show that it needs less number of iterates than the two-grid iteration scheme associated with the discrete collocation method. Numerical results are given which validate the theoretical results.