Conforming and non-conforming virtual element methods for the biharmonic Steklov eigenvalue problem with minimum regularity
Provides a rigorous numerical analysis for a challenging fourth-order eigenvalue problem on nonconvex polygonal domains, extending virtual element methods to this class of problems.
This work develops conforming and C0-non-conforming virtual element methods for the biharmonic Steklov eigenvalue problem on general polygonal domains, proving optimal convergence rates for eigenfunctions and double order for eigenvalues without spurious eigenvalues.
In this work, we analyze the conforming and $C^0$-non-conforming Virtual Element Method for a fourth-order Steklov eigenvalue problem on a generally shaped, possibly nonconvex, polygonal domain. By employing an {\it enriching } operator, we derive the convergence analysis using the discrete $H^2$ seminorm, and the $H^1$ and $L^2$ norms. We use the Babuška--Osborn spectral theory \cite{BO} to prove that the numerical scheme approximates the spectrum without introducing any spurious eigenvalue. Moreover, we derive the optimal order of convergence for eigenfunctions and double order for eigenvalues. We assess the performance of the method on several numerical tests using different families of polygonal meshes.