Adaptive-stabilized finite element methods for eigenvalue problems based on residual minimization onto a dual discontinuous Galerkin norm

Abstract

In this paper, we introduce a framework based on the residual minimization method onto dual discontinuous-Galerkin norms for solving the eigenvalue problem of the Laplace operator. Solving a saddle-point problem allows us to obtain a stable continuous approximation for the eigenfunctions. Furthermore, a residual projection onto a discontinuous polynomial space delivers a robust error estimator for each eigenpair and guides the automatic mesh refinement. Our approach approximates the eigenvalues and eigenfunctions with optimal convergence rates. Finally, numerical results verify our analysis and demonstrate the methodology's excellent performance.