Optimal finite difference grids and rational approximations of the square root. I: Elliptic problems. (English) Zbl 1021.65051

The main objective of this paper is optimization of second-order finite-difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semiinfinite strip is considered. The boundary impedance (Neumann-to-Dirichlet map) is computed as the square root of an operator using the standard three-point finite-difference scheme with optimally chosen variable steps. The finite-difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator’s spectrum.
We implement Zolotarev’s optimal rational approximant obtained in terms of elliptic functions. We also found that a geometrical progression of the grid steps with the optimally chosen parameters is almost as good as the optimal approximant. For bounded operators it increases from second to exponential the convergence order of the finite-difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number. For the case of unbounded operators in Sobolev spaces associated with elliptic equations the error decays as the exponential of the square root of the mesh dimension.
As an example, we numerically compute the Green function on the boundary for the Laplace equation. Some features of the optimal grid obtained for the Laplace equation remain valid for more general elliptic problems with variable coefficients.


65N06 Finite difference methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI


[1] ; , eds. Handbook of mathematical functions, with formulas, graphs and mathematical tables. National Bureau of Standards Applied Mathematics Series, Vol. 55. National Bureau of Standards, Washington, D.C., 1966.
[2] Elements of the theory of elliptic functions. Second revised edition. Izdat. ”Nauka,” Moscow, 1970. Translations of Mathematical Monographs, 79. American Mathematical Society, Providence, R.I., 1990.
[3] Theory of approximation. Reprint of the 1956 English translation. Dover, New York, 1992.
[4] Asvadurov, J Comput Phys 158 pp 116– (2000)
[5] ; Graves-Morris, P. Padé approximants. Second edition. Encyclopedia of Mathematics and Its Applications, 59. Cambridge University, Cambridge, 1996.
[6] Black, SIAM J Sci Comput 19 pp 1667– (1998)
[7] ; Personal communication, 1999.
[8] Spectrally optimal finite difference grids in unbounded domains. Research Note EMG-002-97-22. Schlumberger-Doll Research, Ridgefield, Conn., 1997.
[9] Druskin, SIAM J Matrix Anal Appl 19 pp 755– (1998)
[10] ; Padé, Stieltjes, Lanczos and exponential superconvergence of finite-difference schemes., Proceedings of Copper Mountain Conf. on Iterative Methods. Vol. II. Copper Mountain, Colorado, March-April 1998.
[11] Druskin, SIAM J Numer Anal 37 pp 403– (2000)
[12] Druskin, Numer Algorithms
[13] ; ; On the rate of convergence of Padé approximants of orthogonal expansions. Progress in approximation theory (Tampa, FL, 1990), Springer Ser. Comput. Math. 169-190. 19 Springer, New York, 1992.
[14] Higham, Linear Algebra Appl 88/89 pp 405– (1987)
[15] ; Continued fractions. Analytic theory and applications. Encyclopedia of Mathematics and Its Applications, 11. Addison-Wesley, Reading, Mass., 1980.
[16] ; On the spectral functions of the string. Nine papers in analysis, 19-102. American Mathematical Society Translations, Series 2, Vol. 103. American Mathematical Society, Providence, R.I., 1974. · Zbl 0291.34017
[17] Lu, SIAM J Matrix Anal Appl 19 pp 833– (1998)
[18] Conformal mapping. Reprinting of the 1952 edition. Dover, New York, 1975.
[19] ; Rational approximation of real functions. Encyclopedia of Mathematics and Its Applications, 28. Cambridge University, Cambridge-New York, 1987.
[20] Schmitt, Math Comp 58 pp 191– (1992)
[21] Stahl, Bull Amer Math Soc (NS) 28 pp 116– (1993)
[22] ; Finite element analysis. Wiley, New York, 1991.
[23] Partial differential equations I. Basic theory. Applied Mathematical Sciences, 115. Springer, New York, 1996.
[24] Partial differential equations II. Qualitative studies of linear equations. Applied Mathematical Sciences, 116. Springer, New York, 1996.
[25] Superconvergence in Galerkin finite element methods. Lecture Notes in Mathematics, 1605. Springer Berlin, 1995.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.