This paper proposes two convergent adaptive mesh-refining algorithms for the hybrid high-order method in convex minimization problems with two-sided p-growth. Examples include the p-Laplacian, an optimal design problem in topology optimization, and the convexified double-well problem. The hybrid high-order method utilizes a gradient reconstruction in the space of piecewise Raviart-Thomas finite element functions without stabilization on triangulations into simplices or in the space of piecewise polynomials with stabilization on polytopal meshes. The main results imply the convergence of the energy and, under further convexity properties, of the approximations of the primal resp. dual variable. Numerical experiments illustrate an efficient approximation of singular minimizers and improved convergence rates for higher polynomial degrees. Computer simulations provide striking numerical evidence that an adopted adaptive HHO algorithm can overcome the Lavrentiev gap phenomenon even with empirical higher convergence rates.

翻译：本文为混合高阶方法提出了两种趋同性适应性网格-精密算法,这些算法涉及双向双向双向微增殖的最小化问题,包括p-Laplacian,这是最佳的地形优化设计问题,以及混为一流的双井问题。混合高阶方法利用了小盘拉维特-Thomas有限元素功能空间的梯度重建,而没有稳定地将三角引伸成细微粒或单向多式多式多级计算法空间,在多面藻类中保持稳定。其主要结果意味着能量的趋同,在进一步凝聚性特性下,初线双曲线的近似性能会趋同。数字实验表明单项最小化器的有效接近,提高了高多元度的趋同率。计算机模拟提供了惊人的数字证据,证明一个被采纳的适应性HHHO算法可以克服拉夫伦特耶夫差距现象,即使经验的趋同率更高。