The Virtual Element Method on pixel based domain approximations
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Since its introduction in the early 2010's the virtual element method (VEM) has quickly gained the attention of the scientific community, thanks to its versatility, its robustness (in particular with respect to the shape of the elements), and its potential for high accuracy (the discretization can be designed to be of arbitrarily high order). We analyze and validate its combination with a boundary correction method similar to the shifted boundary method, to solve problems on two and three dimensional domains with curved boundaries, approximated by polygonal/polyhedral domains. We focus on the case of approximate domains obtained as the union of squared elements out of a uniform structured mesh, such as the ones that naturally arise when the continuous domain is issued from an image. We show, both theoretically and numerically, that resorting to polygonal elements allows the assumptions required for stability to be satisfied for any polynomial order. Efficiency is ensured by a novel static condensation strategy acting on the edges/faces of the decomposition. Issues such as the handling of Neumann boundary conditions and the design of the VEM stabilizing term will be addressed.