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.
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