Postprocessing and higher order convergence of the mixed finite element approximations of biharmonic eigenvalue problems (Q557771)

The authors consider the following biharmonic eigenvalue problem: for a given bounded domain \(\varOmega \in \mathbb R^2\) with Lipschitz boundary \(\varGamma\), find \(u(x)\neq 0\) and \( \lambda \in \mathbb R\) satisfying the differential equation \(\Delta^2u(x)=\lambda u(x),\;x\in\varOmega\) subject to homogeneous Dirichlet boundary conditions \(u(x)=0\) and \({\partial u\over \partial \nu}(x)=0,\;x\in \varGamma\), where \( \nu\) is the outward unit normal vector to the boundar \(\varGamma\). A new procedure for accelerating the convergence of mixed finite element approximations of the eigenpairs is proposed. It is based on a postprocessing technique that involves an additional solution of a source problem on a augmented finite element space. This space could be obtained either by substantially refining the grid, the two-grid method, or by using the same grid but increasing the order of polynomials by one, the two-space method. The efficiency of the post-processing algorithm is illustrated on two simple model problems.