Error and stability analysis of boundary methods for elliptic problems with interfaces
Authors:
Zi Cai Li and Rudolf Mathon
Journal:
Math. Comp. 54 (1990), 4161
MSC:
Primary 65N10; Secondary 65N30
MathSciNet review:
990600
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Abstract: In boundary methods, piecewise particular solutions are employed to solve a given elliptic equation within subdomains of some region of interest. A boundary approximation is then obtained by satisfying the interior and exterior boundary conditions in a least squares sense. In this paper, we examine convergence, derive error norm bounds for approximate solutions and conduct a stability analysis of the associated algebraic problem. The aim of this analysis is to help choosing good partitions of subdomains. Finally, numerical experiments are carried out for a typical interface problem, demonstrating that very accurate solutions can be obtained while at the same time keeping small the condition numbers of the associated coefficient matrices.
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 M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs and mathematical tables, Dover, New York, 1980. MR 1225604 (94b:00012)
 [2]
 A. K. Aziz, M. R. Dorr and R. B. Kellogg, A new approximation method for the Helmholtz equation in an exterior domain, SIAM J. Numer. Anal. 19 (1982), 899908. MR 672566 (84d:65080)
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 A. K. Aziz, R. B. Kellogg and A. B. Stephens, Least squares methods for elliptic systems, Math. Comp. 44 (1985), 5370. MR 771030 (86i:65069)
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 F. E. Browder, Approximation by solutions of partial differential equations, Amer. J. Math. 84 (1962), 134160. MR 0178247 (31:2505)
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 S. C. Eisenstat, On the rate of convergence of the BergmanVekua method for the numerical solution of elliptic boundary value problems, SIAM J. Numer. Anal. 11 (1974), 654680. MR 0375805 (51:11995)
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 I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York, 1980.
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 H. Han, The numerical solution of interface problems in finite element methods, Numer. Math. 39 (1982), 3950. MR 664535 (83g:65108)
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 R. B. Kellogg, Singularities in interface problems, in Numerical Solution of Partial Differential Equations II (B. Hubbard, ed.), Academic Press, New York, 1971, pp. 351400. MR 0289923 (44:7108)
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 , A note on Kellogg's eigenfunctions of a periodic SturmLiouville system, Appl. Math. Letters 1 (1988), 123126. MR 953369
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 , A nonconforming combination for elliptic problems with interfaces, J. Comput. Phys. 80 (1989), 288313. MR 1008390 (90h:65183)
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 Z. C. Li and R. Mathon, Boundary approximation methods for solving elliptic problems on unbounded domains, J. Comput. Phys. (to appear). MR 1067051 (91f:65183)
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 Z. C. Li, R. Mathon and P. Sermer, Boundary methods for solving elliptic problems with singularities and interfaces, SIAM J. Numer. Anal. 24 (1987), 487498. MR 888746 (88e:65127)
 [17]
 S. L. Sobolev, Application of functional analysis in mathematical physics, Transl. Math. Monographs, vol. 7, Amer. Math. Soc., Providence, R.I., 1963.
 [18]
 G. Strang and G. J. Fix, An analysis of finite element methods, PrenticeHall, Englewood Cliffs, N.J., 1973. MR 0443377 (56:1747)
 [19]
 R. W. Thatcher, The use of infinite grid refinement at singularities in the solution of Laplace's equation, Numer. Math. 25 (1976), 163178. MR 0400748 (53:4578)
 [20]
 , On the finite element method for unbounded regions, SIAM J. Numer. Anal. 15 (1978), 466477. MR 0471378 (57:11112)
 [21]
 A. N. Tikhonov and A. A. Samarskii, Equations of mathematical physics, Macmillan, New York, 1973. MR 0165209 (29:2498)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199009906004
PII:
S 00255718(1990)09906004
Keywords:
Boundary methods,
elliptic boundary value problem,
interface,
singularity
Article copyright:
© Copyright 1990
American Mathematical Society
