IJPAM: Volume 89, No. 2 (2013)


F.D. d'Almeida$^1$, M. Ahues$^2$, R. Fernandes$^3$
$^1$Centro de Matemática and Faculdade de Engenharia
da Universidade Porto (CMUP)
Rua Roberto Frias, 4200-465 Porto, PORTUGAL
$^2$Institut Camille Jordan
Université Jean Monnet
Membre d'Université de Lyon
23 rue Dr Paul Michelon, 42023 St-Étienne, FRANCE
$^3$Centro de Matemática and Departamento
de Matemática e Aplicações
da Universidade do Minho
Campus de Gualtar, 4710-057 Braga, PORTUGAL

Abstract. In the solution of weakly singular Fredholm integral equations of the second kind defined on the space of Lebesgue integrable complex valued functions by projection methods, the choice of the grid is crucial. We will present the proof of an error bound in terms of the mesh size of the underlying discretization grid on which no regularity assumptions are made and compare it with other recently proposed error bounds. This proof generalizes the work done for the Galerkin method, to the case of Kantorovich and Sloan methods. This allows us to use nonuniform grids when there are boundary layers or discontinuities in the right hand side of the equation. We illustrate this with an example on the radiative transfer model in stellar atmospheres.

Received: August 23, 2013

AMS Subject Classification: 65J10, 65R20

Key Words and Phrases: projection approximations in $L^1$, weakly singular integral operators, nonuniform grids, error bounds

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DOI: 10.12732/ijpam.v89i2.6 How to cite this paper?
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 89
Issue: 2