Boundary Methods for Solving Elliptic Problems with Singularities and Interfaces
Abstract
Boundary approximation techniques are described for solving homogeneous self-adjoint elliptic equations. Piecewise expansions into particular solutions are used which approximate both the boundary and interface conditions in a least squares sense. Convergence of such approximations is proved and error estimates are derived in a natural norm. Numerical experiments are reported for the singular Motz problem which yield extremely accurate solutions with only a modest computational effort.
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References
1.
Stefan Bergman, Integral operators in the theory of linear partial differential equations, Second revised printing. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 23, Springer-Verlag New York Inc., New York, 1969x+145, Berlin
2.
Stanley C. Eisenstat, On the rate of convergence of the Bergman-Vekua method for the numerical solution of elliptic boundary value problems, SIAM J. Numer. Anal., 11 (1974), 654–680
3.
L. Fox, P. Henrici, C. Moler, Approximations and bounds for eigenvalues of elliptic operators, SIAM J. Numer. Anal., 4 (1967), 89–102
4.
L. V. Kantorovich, V. I. Krylov, Approximate methods of higher analysis, Translated from the 3rd Russian edition by C. D. Benster, Interscience Publishers, Inc., New York, 1958xv+681
5.
R. Sherman Lehman, Developments at an analytic corner of solutions of elliptic partial differential equations, J. Math. Mech., 8 (1959), 727–760
6.
Z.-C. Li, An approach for combining the Ritz-Galerkin and finite element methods, J. Approx. Theory, 39 (1983), 132–152
7.
Rudolf Mathon, Pavol Sermer, Numerical solution of the Helmholtz equation, Congr. Numer., 34 (1982), 313–330
8.
R. Mathon, P. Sermer, Convergence of boundary methods for elliptic differential equations, to appear
9.
S. N. Mergelyan, Uniform approximations to functions of a complex variable, Amer. Math. Soc. Transl. Ser. 1, 3 (1962), 294–391
10.
S. L. Sobolev, Applications of functional analysis in mathematical physics, Translated from the Russian by F. E. Browder. Translations of Mathematical Monographs, Vol. 7, American Mathematical Society, Providence, R.I., 1963vii+239
11.
Gilbert Strang, George J. Fix, An analysis of the finite element method, Prentice-Hall Inc., Englewood Cliffs, N. J., 1973xiv+306
12.
R. W. Thatcher, The use of infinite grid refinements at singularities in the solution of Laplace's equation, Numer. Math., 25 (1975/76), 163–178
13.
I. N. Vekua, New methods for solving elliptic equations, Translated from the Russian by D. E. Brown. Translation edited by A. B. Tayler, North-Holland Publishing Co., Amsterdam, 1967xii+358, New York
14.
J. R. Whiteman, N. Papamichael, Treatment of harmonic mixed boundary problems by conformal transformation methods, Math. Phys. Stud., 23 (1972), 655–664
15.
Neil M. Wigley, Asymptotic expansions at a corner of solutions of mixed boundary value problems, J. Math. Mech., 13 (1964), 549–576
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Copyright © 1987 Society for Industrial and Applied Mathematics.
History
Submitted: 12 August 1985
Accepted: 21 July 1986
Published online: 14 July 2006
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