Error estimates for a vectorial second-order elliptic eigenproblem by the local \(L^2\) projected \(C^0\) finite element method (Q2845608)

The paper deals with the \(C^0\) finite element method (FEM) for solving the following vectorial second-order elliptic eigenproblem: Find \((\omega^2, u)\) such that NEWLINE\[NEWLINE\alpha \nabla \times \nabla \times u-\beta \nabla \nabla \cdot u=\omega^2 u \quad \text{in} \quad \Omega \in \mathbb{R}^d,~ d=2,3,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu \times {\mathbf n}=0, \quad \nabla \cdot u=0 \quad \text{on} \quad \Gamma,NEWLINE\]NEWLINE where \(\omega^2 \in \mathbb{R}\), \(u\neq0\), is the unknown eigenpair, \(\alpha, \beta\) are given positive constants and \(\nabla \) is the gradient operator. It is known that e.g. for nonsmooth domains \(\Omega\) with reentrant corners and edges some eigenfunctions can be singular and belong to \((H^r(\Omega))^d\) for some real \(r<1\) only which causes the wrong convergence of the FEM to an element outside \((H^r(\Omega))^d\). It is shown that the method proposed guarantees the convergence order \({\mathcal O}(h^{2r})\) for the eigenvalues and \({\mathcal O}(h^r)\) for the singular eigenfunctions. The crutial tools for the theoretical framework including the optimal error bounds for singular solutions in \(H^r\), \(r>1/2\), are Fortin-type interpolations and an inf-sup inequality.