Accuracy of the difference scheme of solving the eigenvalue problem for the Laplacian
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ACCURACY OF THE DIFFERENCE SCHEME OF SOLVING THE EIGENVALUE PROBLEM FOR THE LAPLACIAN N. V. Maiko,a† V. G. Prikazchikov,a and V. L. Ryabicheva‡
UDC 519.6
Abstract. The finite-difference approximation of the eigenvalue problem with the Dirichlet boundary conditions for the Laplacian in a two-dimensional domain of complex form is analyzed for accuracy and the error of eigenfunctions from the class W22 (W ) in the mesh norm of W21 ( w ) is estimated. Keywords: Laplacian, eigenvalues, difference scheme, rate of convergence.
We will perform the accuracy analysis of the difference approximation of an eigenvalue problem with Dirichlet boundary conditions for a Laplacian in a two-dimensional domain of complex form. The scientific literature describes various discrete analogs of a Laplacian; however, not all of them are self-adjoint difference operators (for example, see [1, p. 241]). In our case, we will use the same approximation of a Laplacian on a five-point template of a nonuniform mesh as in [2], leading to a self-adjoint difference operator. The paper [2] performed the accuracy analysis of the difference scheme of the solution of the first boundary-value problem for an elliptic equation with variable coefficients in a two-dimensional domain W of complex form and obtained the estimates || y - u ||C ( w ) £ Mh 2 , | | y - u | |W 1 ( w ) £ Mh 3 / 2 2
on the assumption that u Î C 4 ( W ) , where u and y are the solutions of the differential and difference problems, respectively, and M is a positive constant independent of step h. In [3], the difference scheme of the solution of the same problem was analyzed on the assumption that u Î W2m (W ) for m = 2, 3 and the rate of convergence was estimated: || y - u ||W 1 ( w ) £ Mh m / 2 | | u ||W m (W ) 2
2
æ h = h2 + h2 ö . ç 1 2 ÷ø è
The purpose of our study is to obtain the estimate of the same type provided that the generalized eigenfunctions belong to the class W22 (W ) . 1. Consider an eigenvalue problem -Du º -
¶2u ¶x12
-
¶2u ¶x 22
= lu, x = ( x1 , x 2 ) Î W,
u = 0, x = ( x1 , x 2 ) Î G º ¶W.
(1)
(2)
o
Recall that a nonzero function u Î W21 (W ) is called the generalized eigenfunction of the first boundary-value problem, or the Dirichlet problem, for the operator L = -D if there exists a number l such that the function u satisfies the following a
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, †[email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 131–139, September–October 2011. Original article submitted May 17, 2010. 1060-0396/11/4705-0783
©
2011 Springer Science+Business Media, Inc.
783
o
integral identity for all v Î W21 (W ) :
2
¶u ¶v dx = l òò uvdx. ¶x a
òò å ¶xa W a =1
W
The number l is called the eigenvalue that corresponds to the eigenfunction u, which is assumed normalized, for æ ö1/ 2 example, by the condition || u ||L2 ( w ) º ç òò u 2 ( x )dx ÷ = 1 . ç ÷ èW ø As is known from [4, 5], problem (1), (2) has a countable set of positive eigenvalues: 0 < l 1 £ l 2 £ K £ l n
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