The hybrid high-order method is a modern numerical framework for the approximation of elliptic PDEs. We present here an extension of the hybrid high-order method to meshes possessing curved edges/faces. Such an extension allows us to enforce boundary conditions exactly on curved domains, and capture curved geometries that appear internally in the domain e.g. discontinuities in a diffusion coefficient. The method makes use of non-polynomial functions on the curved faces and does not require any mappings between reference elements/faces. Such an approach does not require the faces to be polynomial, and has a strict upper bound on the number of degrees of freedom on a curved face for a given polynomial degree. Moreover, this approach of enriching the space of unknowns on the curved faces with non-polynomial functions should extend naturally to other polytopal methods. We show the method to be stable and consistent on curved meshes and derive optimal error estimates in $L^2$ and energy norms. We present numerical examples of the method on a domain with curved boundary, and for a diffusion problem such that the diffusion tensor is discontinuous along a curved arc.

翻译：杂交高阶方法是一种求解椭圆偏微分方程的现代数值框架。本文提出将该方法推广至具有弯曲边/面的网格。这一推广能够精确在弯曲域上施加边界条件，并捕捉域内出现的弯曲几何特征（例如扩散系数的不连续性）。该方法在弯曲面上采用非多项式函数，无需引入参考单元/面的映射。此方法不要求弯曲面为多项式曲面，且在给定多项式阶数下，弯曲面上的自由度数量存在严格上界。此外，这种通过非多项式函数丰富弯曲面未知量空间的方法应能自然推广至其他多面体方法。我们证明该方法在弯曲网格上具有稳定性和相容性，并推导出$L^2$范数与能量范数下的最优误差估计。最后给出数值算例，包括具有弯曲边界的域以及扩散张量沿弯曲弧不连续的问题。