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

Vanmaele, M.

Simon Stevin 67, No. 1-2, 121-132 (1993).

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 \(1/2\leq \varepsilon< 1\).

Reviewer: P.Chocholatý (Bratislava)

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

Keywords:

optimal error estimates; numerical quadrature finite element method; second order elliptic eigenvalue problems; Lobatte quadrature formula; Gauss-Legendre quadrature formula; boundary value problem; eigenfunctions