A priori and a posteriori error analysis for discontinuous Galerkin finite element approximations of biharmonic eigenvalue problems

In this paper, we express and analyze mixed discontinuous Galerkin(DG) methods of biharmonic eigenvalue problems as well as present the error analysis for them. The analysis consists of two parts. First, we propose a residual-based a posteriori error estimator in the approximate eigenfunctions and eigenvalues. The error in the eigenfunctions is measured both in the L² and DG (energy-like) norms. In addition, we prove that if the error estimator converges to zero, then the distance of the computed eigenfunction from the true eigenspace also converges to zero, and so, the computed eigenvalue converges to a true eigenvalue. Next, we establish an a priori error estimate with the optimal convergence order both in the L² and DG norms. We show that the methods can retain the same convergence properties they enjoy in the case of source problems.