Convergence and Error Estimates for a Finite Element Method with Numerical Quadrature for a Second Order Elliptic Eigenvalue Problem

Abstract

This paper deals with a FE-numerical quadrature method for a class of 2nd order elliptic eigenvalue problems on a bounded rectangular domain Ω ⊂ ℝ², viz.

$$ {\text{Find }}\lambda {\text{ }} \in {\text{ }}\mathbb{R}{\text{, u }} \in {\text{ V : a(u,v) = }}\lambda {\text{. (u,v) }}\forall {\text{ v }} \in {\text{ V}} $$

((1.1))

where

$$ \begin{array}{*{20}{c}} {{\text{V}}\{ {\text{v}} \in {\text{ }}{{{\text{H}}}^{1}}{\text{(}}\Omega ){\text{|v = 0on}}{{\Gamma }_{1}}{\text{ }} \subset {\text{ }}\partial \Omega {\text{ = boundary of }}\Omega \} } \\ {{\text{(}}{{\Gamma }_{1}}{\text{ consisting of an integer number of sides)}}} \\ {(.,.) = {{{\text{L}}}_{{\text{2}}}}{\text{ (}}\Omega {\text{) - inner product}}} \\ {{\text{a(u,v) = }}\int\limits_{\Omega } {[\sum\nolimits_{{{\text{i,j = l}}}}^{2} {{{{\text{a}}}_{{{\text{ij}}}}}{\text{(x)}}\cdot \frac{{\partial {\text{u}}}}{{\partial {{{\text{x}}}_{{\text{i}}}}}}\cdot \frac{{\partial {\text{v}}}}{{\partial {{{\text{x}}}_{{\text{j}}}}}}{\text{ + }}{{{\text{a}}}_{0}}{\text{(x)}}{\text{.u}}.{\text{v}}]{\text{.dx, x = (}}{{{\text{x}}}_{1}}{\text{,}}{{{\text{x}}}_{2}}{\text{)}}{\text{.}}} } } \\ \end{array} $$