论文标题
一种在混合高阶方法中处理弯曲网格的新方法
A new approach to handle curved meshes in the hybrid high-order method
论文作者
论文摘要
混合高阶方法是用于近似椭圆PDE的现代数值框架。我们在这里向具有弯曲边缘/面的网格介绍了混合高阶方法的扩展。这样的扩展使我们能够在弯曲域上精确地执行边界条件，并捕获内部出现在域中的弯曲几何形状，例如扩散系数中的不连续性。该方法利用在弯曲的面上使用非多功能功能，并且不需要参考元素/面之间的任何映射。这种方法不需要面部是多项式的，并且在给定的多项式程度上，在弯曲面上的自由度上具有严格的上限。此外，这种富集非物质函数弯曲面上未知数空间的方法应自然扩展到其他多面有方法。我们展示了在弯曲的网格上保持稳定且一致的方法，并在$ l^2 $和能量规范中得出最佳误差估计。我们介绍了该方法在具有弯曲边界的域上的数值示例，并且对于扩散问题，因此扩散张量沿弯曲的弧不连续。
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.
