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On optimal and nearly optimal error estimates of a numerical quadrature finite element method for 2nd-order eigenvalue problems with Dirichlet boundary conditions. (English) Zbl 0802.65106
This paper deals with a numerical quadrature finite elements (FE) method for a class of second order elliptic eigenvalue problems on a rectangular domain $$\Omega\subset\mathbb{R}^ 2$$ with homogeneous Dirichlet boundary conditions on the whole boundary $$\partial\Omega$$, in the form (1) find $$(\lambda u)\in\mathbb{R}\times H^ 1_ 0(\Omega)$$: $$a(u,v)= \lambda(u,v)$$, $$\forall v\in V\equiv H^ 1_ 0(\Omega)$$.
The author considers the numerical quadrature FE-approximation of (1) and obtains results for the approximate eigenpairs $$(\bar\lambda_{e,h}, \bar u_{e,h})$$. In the right-hand side a discrete inner product $$(\cdot,\cdot)_ h$$ is used, corresponding to a suitable Lobatto quadrature formula for the involved integrals on each element $$K$$. In the left-hand side the approximation $$a_ h(\cdot,\cdot)$$ of the bilinear form $$a(\cdot,\cdot)$$ results from a deliberately chosen Gauss-Legendre quadrature formula for the involved integrals on the elements $$K$$.
When the boundary value problem associated with the bilinear form is ‘regular’, the author improves the estimate for the eigenfunctions in the $$L_ 2(\Omega)$$-norm to a nearly optimal one, namely $$O(h^{k+\varepsilon})$$ with $$1/2\leq \varepsilon< 1$$.

##### MSC:
 65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35P15 Estimates of eigenvalues in context of PDEs