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arXiv · 2606.15340

Conforming and non-conforming virtual element methods for the biharmonic Steklov eigenvalue problem with minimum regularity

Dibyendu Adak·Daniele Boffi·Francesca Gardini·Gianmarco Manzini·Jesus Vellojin
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Abstract

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 Babu\v{s}ka--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.

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Dibyendu Adak, Daniele Boffi, Francesca Gardini, Gianmarco Manzini, Jesus Vellojin. 2026-06-13. Conforming and non-conforming virtual element methods for the biharmonic Steklov eigenvalue problem with minimum regularity. https://arxiv.org/abs/2606.15340

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