We investigate the convergence properties of a mixed finite element method approximation to the Biharmonic eigenvalue problem under the presence of numerical integration. We give a brief overview of the results obtained when exact integration is used in a finite element method, then develop related theories and obtained the convergence rates when numerical quadrature is taken into account. The standard approach to obtaining error estimate of variational eigenvalue problems is based on the error estimate of the solution operators of the source problems. The important issues are the rate of convergence of the solution operators and the conditions required for convergence. Paralleling the work of Babuska, Osborn and Pikaranta to overcome some technical difficulties, we will use mesh dependent norms to obtain error estimates between the solutions operators. We then use these estimates to get errors estimates between the approximate and the actual eigenvalues and eigenvectors.
Tran, Max, "Finite Element Spectral Approximation with Numerical Integration for the Biharmonic Eigenvalue Problem" (2014). CUNY Academic Works.