The role of mesh quality and mesh quality indicators in the virtual element method

• Published in: Advances in computational mathematics Vol. 48; no. 1
• Main Authors: Sorgente, T., Biasotti, S., Manzini, G., Spagnuolo, M.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2022; Springer Nature B.V
• Subjects: Computational mathematics; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Convergence; Finite element method; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Norms; Optimization; Regularity; Visualization; Mesh quality indicators; 65N12; Mesh regularity assumptions; 2D Poisson problem; Polygonal mesh; 65N50; Optimal convergence; 68U05; Virtual element method; 65N30; Small edges
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-021-09913-3

Since its introduction, the virtual element method (VEM) was shown to be able to deal with a large variety of polygons, while achieving good convergence rates. The regularity assumptions proposed in the VEM literature to guarantee the convergence on a theoretical basis are therefore quite general. They have been deduced in analogy to the similar conditions developed in the finite element method (FEM) analysis. In this work, we experimentally show that the VEM still converges, with almost optimal rates and low errors in the L 2 , H 1 and L ∞ norms, even if we significantly break the regularity assumptions that are used in the literature. These results suggest that the regularity assumptions proposed so far might be overestimated. We also exhibit examples on which the VEM sub-optimally converges or diverges. Finally, we introduce a mesh quality indicator that experimentally correlates the entity of the violation of the regularity assumptions and the performance of the VEM solution, thus predicting if a mesh is potentially critical for VEM.