3D Adaptive VEM with Stabilization-Free a Posteriori Error Bounds

Abstract The present paper extends the theory of Adaptive Virtual Element Methods (AVEMs) started in (Beirão da Veiga et al. in SIAM J Numer Anal 61(2):457–494, https://doi.org/10.1137/21M1458740 , 2023) to the three-dimensional meshes showing the possibility to bound the stabilization term by the residual-type error estimator. This new bound establishes the equivalence between a stabilization-free residual-type a posteriori error estimator and the energy error, enabling the formulation of a 3D AVEM algorithm and providing the necessary results to prove its convergence. Following the recent studies for the bi-dimensional case, we investigate the case of tetrahedral elements with aligned edges and faces. We believe that the AVEMs can be an efficient strategy to address the mesh conforming requirements of standard three-dimensional Adaptive Finite Element Methods (AFEMs), which typically extend the refinement procedure to non-marked mesh cells. Indeed, numerical tests show that this method can reduce the number of three-dimensional cells generated in the refinement process up to $$30\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>30</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> with respect to standard AFEMs, for a given error threshold.