IJPAM: Volume 55, No. 4 (2009)

PIECEWISE CONSTANT GALERKIN APPROXIMATIONS
OF WEAKLY SINGULAR INTEGRAL EQUATIONS

M. Ahues$^1$, F.D. d'Almeida$^2$, R.R. Fernandes$^3$
$^1$Laboratoire de Mathématiques
l'Université de St-Étienne
Membre d'Université de Lyon, EA3989
23, Rue Dr Paul Michelon, St-Étienne, 42023, FRANCE
e-mail: mario.ahue@univ-st-etienne.fr
$^2$Centro de Matemática
Faculdade Engenharia
Universidade Porto -- CMUP
Rua Roberto Frias, Porto, 4200-465, PORTUGAL
e-mail: falmeida@fe.up.pt
$^3$Centro and Departamento de Matemática
Universidade do Minho
Campus de Gualtar, Braga, 4710-057, PORTUGAL
e-mail: rosario@math.uminho.pt


Abstract.We consider a weakly singular 2-nd kind Fredholm integral equation defined on the space of Lebesgue integrable complex-valued functions. From all standard projection approximations of a bounded linear operator in a Banach space, the Galerkin scheme is the simplest one from a computational point of view. We explore its rate of convergence in terms of the mesh size of the underlying discretization grid on which no regularity assumption is made. An example in Astrophysics illustrates the actual behaviour of the error in terms of the distribution of the points in the grid.

Received: August 17, 2009

AMS Subject Classification: 65J10, 65R20

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

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 55
Issue: 4