Published

Anisotropic Error Estimates of The Linear Virtual Element Method on Polygonal Meshes

Shuhao Cao and Long Chen

SIAM Journal on Numerical Analysis, 56(5), 2913-2939, 2018

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ABSTRACT:

 A refined a priori error analysis of the lowest order
(linear) Virtual Element Method (VEM) is developed for approximating a
model two dimensional Poisson problem. A set of new geometric
assumptions is proposed on shape regularity of polygonal meshes. A new
universal error equation for the lowest order (linear) VEM is derived
for any choice of stabilization, and a new stabilization using broken
half-seminorm is introduced to incorporate short edges naturally into
the a priori error analysis on isotropic elements. The error analysis
is then extended to a special class of anisotropic elements with high
aspect ratio originating from a body-fitted mesh generator, which uses
straight lines to cut a shape regular background mesh. Lastly, some
commonly used tools for triangular elements are revisited for
polygonal elements to give an in-depth view of these estimates'
dependence on shapes.