Convergent adaptive hybrid higher-order schemes for convex minimization

TL;DR

This paper introduces two adaptive mesh-refining algorithms for hybrid high-order methods, demonstrating convergence and improved accuracy in convex minimization problems with complex features like singular minimizers.

Contribution

It develops novel adaptive algorithms for hybrid high-order methods, ensuring convergence and enhanced performance in convex minimization tasks with complex geometries.

Findings

Algorithms converge for energy and variables

Adaptive method overcomes Lavrentiev gap phenomenon

Higher polynomial degrees yield faster convergence

Abstract

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.