The combined effect of numerical integration and approximation of the boundary in the finite element method for eigenvalue problems

Abstract

The paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains\(\Omega _h\) are not subdomains of the original domain\(\Omega \subset \R^2\) and when at the same time numerical integration is used for computing the involved bilinear forms. The considerations are restricted to piecewise linear approximations. The optimum rate of convergence\(O (h^2)\) for approximate eigenvalues is obtained provided that a quadrature formula of first degree of precision is used. In the case of a simple exact eigenvalue the optimum rate of convergence\(O (h)\) for approximate eigenfunctions in the\(H^1 (\Omega _h)\) -norm is proved while in the\(L_2 (\Omega _h)\) -norm an almost optimum rate of convergence (i.e. near to\(O (h^2))\) is achieved. In both cases a quadrature formula of first degree of precision is used. Quadrature formulas with degree of precision equal to zero are also analyzed and in the case when the exact eigenfunctions belong only to\(H^1(\Omega )\) the convergence without the rate of convergence is proved. In the case of a multiple exact eigenvalue the approximate eigenfunctions are compard (in contrast to standard considerations) with linear combinations of exact eigenfunctions with coefficients not depending on the mesh parameter \(h\).