An adjoint-based super-convergent Galerkin approximation of eigenvalues

We present a new method for computing high-order accurate approximations of eigenvalues defined in terms of Galerkin approximations. We consider the eigenvalue as a non-linear functional of its corresponding eigenfunction and show how to extend the adjoint-based approach proposed in Cockburn and Wang (2017) [14], to compute it. We illustrate the method on a second-order elliptic eigenvalue problem. Our extensive numerical results show that the approximate eigenvalues computed by our method converge with a rate of 4k+2 when tensor-product polynomials of degree k are used for the Galerkin approximations. In contrast, eigenvalues obtained by standard finite element methods such as the mixed method or the discontinuous Galerkin method converge with a rate at most of 2k+2. We present numerical results which show the performance of the method for the classic unit square and L-shaped domains, and for the quantum harmonic oscillator. We also present experiments uncovering a new adjoint-corrected approximation of the eigenvalues provided by the hybridizable discontinuous Galerkin method which converges with order 2k+2, as well as results showing the possibilities and limitations of using the adjoint-correction term as an asymptotically exact a posteriori error estimate.