We analyze and validate the virtual element method combined with a boundary correction similar to the one in [1,2], to solve problems on two dimensional domains with curved boundaries approximated by polygonal domains obtained as the union of squared elements out of a uniform structured mesh, such as the one that naturally arises when the domain is issued from an image. We show, both theoretically and numerically, that resorting to the use of polygonal elements allows to satisfy, for any order, the assumptions required for the stability of the method, thus allowing to fully exploit the potential of higher order methods, the efficiency of which is ensured by a novel static condensation strategy acting on the edges of the decomposition.

翻译：我们分析并验证了结合边界修正的虚拟元方法（修正方法类似于文献[1,2]），用于求解二维区域上具有曲线边界的问题。该区域通过均匀结构化网格中正方形单元并集形成的多边形域近似，例如从图像生成区域时自然产生的网格类型。我们从理论和数值两方面证明，采用多边形单元可以在任意阶数下满足方法稳定性所需假设，从而充分发掘高阶方法的潜力，其效率通过一种作用于剖分边界的创新静态凝聚策略得以保证。