We present both $hp$-a priori and $hp$-a posteriori error analysis of a mixed-order hybrid high-order (HHO) method to approximate second-order elliptic problems on simplicial meshes. Our main result on the $hp$-a priori error analysis is a $\frac12$-order $p$-suboptimal error estimate. This result is, to our knowledge, the first of this kind for hybrid nonconforming methods and matches the state-of-the-art for other nonconforming methods as discontinuous Galerkin methods. Our second main result is a residual-based $hp$-a posteriori upper error bound, comprising residual, normal flux jump, tangential jump, and stabilization estimators (plus data oscillation terms). The first three terms are $p$-optimal and only the latter is $\frac12$-order $p$-suboptimal. This result is, to our knowledge, the first $hp$-a posteriori error estimate for HHO methods. A novel approach based on the partition-of-unity provided by hat basis functions and on local Helmholtz decompositions on vertex stars is devised to estimate the nonconformity error. Finally, we establish local lower error bounds. Remarkably, the normal flux jump estimator is only $\frac12$-order $p$-suboptimal, as it can be bounded by the stabilization owing to the local conservation property of HHO methods. Numerical examples illustrate the theory.

翻译：本文针对单纯形网格上的二阶椭圆问题，提出了一种混合阶杂交高阶（HHO）方法的$hp$先验与$hp$后验误差分析。关于$hp$先验误差分析的主要结果是$\frac12$阶$p$次优误差估计。据我们所知，这是杂交非协调方法中首个此类结果，且与不连续伽辽金方法等其他非协调方法的最先进成果相匹配。我们的第二个主要结果是基于残差的$hp$后验误差上界，包含残差、法向通量跳跃、切向跳跃及稳定化估计量（以及数据振荡项）。前三项具有$p$最优性，仅最后一项为$\frac12$阶$p$次优。据我们所知，这是HHO方法中首个$hp$后验误差估计。为估计非协调误差，我们设计了一种基于帽基函数提供的单位分解及顶点星形域上局部亥姆霍兹分解的新方法。最后，我们建立了局部误差下界。值得注意的是，由于HHO方法的局部守恒特性，法向通量跳跃估计量仅为$\frac12$阶$p$次优，因其可被稳定化项控制。数值算例验证了理论结果。