Convergent adaptive hybrid higher-order schemes for convex minimization

TL;DR

This work addresses convex minimization problems with two-sided $p$-growth on a bounded domain, where standard discretizations may fail to converge reliably or capture singular minimizers due to Lavrentiev gaps. It introduces adaptive hybrid high-order methods (AHHO) with and without stabilization, employing gradient reconstructions into Raviart–Thomas spaces and potential reconstructions to drive an a posteriori, residual-free refinement indicator and Dörfler marking. The authors prove plain convergence: the discrete energies converge to the exact minimum, and under convexity-type conditions, the discrete gradients and stresses converge to their continuous counterparts; the stabilized version extends to polytopal meshes. Numerical experiments on $p$-Laplacian, optimal design, and relaxed two-well problems show that AHHO can efficiently approximate singular minimizers, achieve higher convergence rates with increasing polynomial degree, and empirically overcome the Lavrentiev gap. The results thus provide a robust, convergent framework for adaptive nonconforming discretizations of nonlinear convex problems, with potential 3D extensions and broad applicability to topology optimization and phase-transition models.

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.

Figures (11)

• Figure 1: Polynomial degrees $k = 0,\dots,4$ in the numerical benchmarks of \ref{['sec:numerical-examples']}
• Figure 2: Initial triangulation $\mathcal{T}_0$ (left) of the L-shaped domain and convergence history plot (right) of $|E(u) - E_\ell(u_\ell)|$ with $k$ from \ref{['fig:legend']} on uniform (dashed line) and adaptive (solid line) triangulations for the $p$-Laplace benchmark in \ref{['sec:num-ex:p-Laplace']}
• Figure 3: Adaptive triangulations of the L-shaped domain into 492 triangles (1238 dofs) for $k = 0$ (left) and 490 triangles (7824 dofs) for $k = 3$ (right) for the $p$-Laplace benchmark in \ref{['sec:num-ex:p-Laplace']}
• Figure 4: Convergence history plot of $\|\nabla u - \mathcal{G}_\ell u_\ell\|_{L^4(\Omega)}^2$ (left) and $\|\sigma - \nabla W(\mathcal{G}_\ell u_\ell)\|_{L^{4/3}(\Omega)}^2$ (right) with $k$ from \ref{['fig:legend']} on uniform (dashed line) and adaptive (solid line) triangulations for the $p$-Laplace benchmark in \ref{['sec:num-ex:p-Laplace']}
• Figure 5: Material distribution of the L-shaped domain (left) and convergence history plot (right) of $\mathrm{RHS}_\ell$ in \ref{['ineq:RHS']} with $k$ from \ref{['fig:legend']} on uniform (dashed line) and adaptive (solid line) triangulations for the optimal design problem in \ref{['sec:num-ex:ODP']}

Theorems & Definitions (31)

• Theorem 2.1: plain convergence
• Theorem 2.2: plain convergence for stabilized HHO
• Lemma 3.1: properties of $\mathcal{G}$
• proof
• Theorem 3.2: discrete minimizers
• proof
• Remark 3.3: $H(\mathrm{div})$ conformity
• Lemma 3.4: right-inverse
• proof
• Theorem 4.1: discrete compactness