THE COMBINED EFFECT OF NUMERICAL-INTEGRATION AND APPROXIMATION OF THEBOUNDARY IN THE FINITE-ELEMENT METHOD FOR EIGENVALUE PROBLEMS

The paper deals with the finite element analysis of second order ellip tic eigenvalue problems when the approximate domains Omega(h) are not subdomains of the original domain Omega subset of R(2) and when at the same time numerical integration is used for computing the involved bi linear forms. The considerations are restricted to piecewise linear ap proximations, The optimum rate of convergence O(h(2)) for approximate eigenvalues is obtained provided that a quadrature formula of first de gree of precision is used. In the case of a simple exact eigenvalue th e optimum rate of convergence O(h) for approximate eigenfunctions in t he H-1(Omega(h))-norm is proved while in the L(2)(Omega(h))-norm an al most optimum rate of convergence (i.e. near to O(h(2))) is achieved, I n 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 eigenfuncti ons are compard (in contrast to standard considerations) with linear c ombinations of exact eigenfunctions with coefficients not depending on the mesh parameter h.