A superconvergence result for mixed finite element approximations of the eigenvalue problem (Q2842466)

The paper deals with the following second-order elliptic eigenvalue problem: find \(\left( p,\lambda \right) \;\)such that NEWLINE\[NEWLINE -\nabla \cdot \left( \mathcal{A\nabla }p\right) +\varphi p=\lambda \rho p \text{ in }\Omega ,\text{ }\mathcal{B}\left( p\right) =0\text{ on }\partial \Omega ,\text{ }\int_{\partial \Omega }\rho p^{2}\text{d}\Omega =1,\; NEWLINE\]NEWLINE \ where\ \(0\leq \varphi \in W^{0,\infty }\left( \Omega \right)\), \( 0<c_{0}\leq \rho \in W^{0,\infty }\left( \Omega \right)\), \(\Omega \subset \mathbb{R}^{2}\) is a bounded domain with Lipschitz boundary \(\partial \Omega \), \(\mathcal{A}\) is a symmetric positive definite matrix with the components in \(W^{1,\infty }\left( \Omega \right) \) and \(\mathcal{B}\left( p\right) \) denotes the boundary condition which can be of Dirichlet or Neumann type. The authors prove the superconvergence between the eigenfunction approximation and its corresponding mixed finite element projection for the lowest-order Raviart-Thomas approximation.They introduce a new way to derive the superconvergence by general mixed finite element methods which have the commuting diagram property. Numerical experiments confirm the theoretical analysis.