Abstract
Some error analyses on virtual element methods (VEMs) including inverse inequalities, norm equivalence, and interpolation error estimates are developed for polygonal meshes, each element of which admits a virtual quasi-uniform triangulation. This sub-mesh regularity covers the usual ones used for theoretical analysis of VEMs, and the proofs are presented by means of standard technical tools in finite element methods.
Highlights
Since the pioneer work in [2, 4, 5], virtual element methods (VEMs) have widely been used for numerically solving various partial differential equations in recent years
As an application of the inverse inequality, we prove the L2-stability of the projection QK and Π∇k restricted to VEM spaces
We note that A2 will rule out polygons with high aspect ratio
Summary
Since the pioneer work in [2, 4, 5], virtual element methods (VEMs) have widely been used for numerically solving various partial differential equations in recent years. If we apply the generalized scaling argument to derive the estimate (1) for virtual element spaces, since VK is defined with the help of the Laplacian operator (for details see [2, 4] or Section 2), we require to show the solution of the Poisson equation defined over K depends on the shape of K continuously. Such results may be obtained rigorously in a very subtle and technical way. Norm equivalence, and interpolation error estimates for several types of VEM spaces are derived technically in Sections 3-5, respectively
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