Anhp-Hybrid High-Order Method for Variable Diffusion on General Meshes

Abstract

AbstractIn this work, we introduce and analyze anhp-hybrid high-order (hp-HHO) method for a variable diffusion problem. The proposed method is valid in arbitrary space dimension and for fairly general polytopal meshes. Variable approximation degrees are also supported. We provehp-convergence estimates for both the energy- andL2L^{2}-norms of the error, which are the first of this kind for Hybrid High-Order methods. These results hinge on a novelhp-approximation lemma valid for general polytopal elements in arbitrary space dimension. The estimates are additionally fully robust with respect to the heterogeneity of the diffusion coefficient, and show only a mild dependence on the square root of the local anisotropy, improving previous results for HHO methods. The expected exponential convergence behavior is numerically demonstrated on a variety of meshes for both isotropic and strongly anisotropic diffusion problems.