Abstract
In this paper we propose a fully discretized version of the collocation method applied to integral equations of the first kind with logarithmic kernel. After a stability and convergence analysis is given, we prove the existence of an asymptotic expansion of the error, which justifies the use of Richardson extrapolation. We further show how these expansions can be translated to a new expansion of potentials calculated with the numerical solution of a boundary integral equation such as those treated before. Some numerical experiments, confirming our theoretical results, are given.
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References
M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
D. Arnold and W.L. Wendland, The convergence of spline collocation for strongly elliptic equations on curves, Numer. Math. 47 (1985) 317–341.
R. Celorrio and F.-J. Sayas, Extrapolation techniques and the collocation method for a class of boundary integral equations, to appear in J. Austral. Math. Soc. Ser. B.
R. Celorrio and F.-J. Sayas, The Euler–Maclaurin formula in presence of a logarithmic singularity, BIT 39 (1999) 780–785.
M. Crouzeix and F.-J. Sayas, Asymptotic expansions of the error of spline Galerkin boundary element methods, Numer. Math. 78 (1998) 523–547.
G.C. Hsiao, P. Kopp and W.L. Wendland, A Galerkin collocation method for some integral equations of the first kind, Computing 25 (1980) 89–130.
W. McLean, Fully-discrete collocation methods for an integral equation of the first kind, J. Integral Equations Appl. 6 (1994) 537–571.
W. Rudin, Functional Analysis (MacGraw-Hill, New York, 1973).
J. Saranen, The convergence of even degree spline collocation solution for potential problems in smooth domains of the plane, Numer. Math. 53 (1988) 499–512.
J. Saranen, On the effect of numerical quadrature in solving boundary integral equations, in: Notes on Numerical Fluid Mechanics, Vol. 21 (Vieweg, Braunschweig, 1988) pp. 196–209.
J. Saranen and W.L. Wendland, On the asymptotic convergence of collocation methods with spline functions of even degree, Math. Comp. 45 (1985) 91–108.
F.-J. Sayas, Asymptotic expansions of the error of some boundary element methods, Thesis doctoral, University of Zaragoza (1994).
F.-J. Sayas, Fully discrete Galerkin methods for systems of boundary integral equations, J. Comput. Appl. Math. 81 (1997) 311–331.
I.H. Sloan, Boundary element methods, in: Theory and Numerics of Ordinary and Partial Differential Equations, eds. M. Ainsworth et al. (Oxford Science, Oxford, 1995) pp. 138–180.
J.F. Steffensen, Interpolation, 2nd ed. (Chelsea, New York, 1950).
J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer, Berlin, 1972).
Y. Yan and I.H. Sloan, On integral equations of the first kind with logarithmic kernels, J. Integral Equations Appl. 1 (1988) 549–579.
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Celorrio, R., Sayas, FJ. Full collocation methods for some boundary integral equations. Numerical Algorithms 22, 327–351 (1999). https://doi.org/10.1023/A:1019127428490
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DOI: https://doi.org/10.1023/A:1019127428490