We propose an Extended Hybrid High-Order scheme for the Poisson problem with solution possessing weak singularities. Some general assumptions are stated on the nature of this singularity and the remaining part of the solution. The method is formulated by enriching the local polynomial spaces with appropriate singular functions. Via a detailed error analysis, the method is shown to converge optimally in both discrete and continuous energy norms. Some tests are conducted in two dimensions for singularities arising from irregular geometries in the domain. The numerical simulations illustrate the established error estimates, and show the method to be a significant improvement over a standard Hybrid High-Order method.

我们针对具有弱奇异性的解的 Poisson 问题提出了一种扩展混合高阶方案。对这个奇点的性质和解的剩余部分陈述了一些一般性假设。该方法是通过用适当的奇异函数丰富局部多项式空间来制定的。通过详细的误差分析，该方法被证明在离散和连续能量范数中都以最佳方式收敛。一些测试是在二维中针对由域中不规则几何形状引起的奇点进行的。数值模拟说明了已建立的误差估计，并表明该方法是对标准混合高阶方法的显着改进。