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. We focus on the case of approximating 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 polygonal elements allows the assumptions required for stability to be satisfied for any polynomial order. This allows us to fully exploit the potential of higher order methods. Efficiency 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（类似于文献[1,2]中的方法），用于求解具有曲线边界且由多边形区域近似的二维域问题。我们重点关注通过均匀结构化网格中的方形单元并集所获得的近似域，例如从图像中提取计算域时自然产生的情形。我们从理论和数值两方面证明，采用多边形单元能够满足任意多项式阶数所需的稳定性假设，从而充分发挥高阶方法的潜力。通过一种作用于分解边的新型静态凝聚策略，确保了计算效率。