# Convergent adaptive hybrid higher-order schemes for convex minimization

@article{Carstensen2022ConvergentAH, title={Convergent adaptive hybrid higher-order schemes for convex minimization}, author={Carsten Carstensen and Ngoc Tien Tran}, journal={ArXiv}, year={2022}, volume={abs/2111.01181} }

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…

## References

SHOWING 1-10 OF 60 REFERENCES

An Arbitrary-Order and Compact-Stencil Discretization of Diffusion on General Meshes Based on Local Reconstruction Operators

- Computer ScienceComput. Methods Appl. Math.
- 2014

An arbitrary-order primal method for diffusion problems on general polyhedral meshes based on a local (elementwise) discrete gradient reconstruction operator that is proved to optimally converge in the energy norm and in the L2-norm of the potential for smooth solutions.

A convergent adaptive finite element method for an optimal design problem

- MathematicsNumerische Mathematik
- 2008

This work motivates a simple edge-based adaptive mesh-refining algorithm (AFEM) that is not a priori guaranteed to refine everywhere and its convergence proof is therefore based on energy estimates and some refined convexity control.

Quasi-Optimal Convergence Rate of an Adaptive Discontinuous Galerkin Method

- MathematicsSIAM J. Numer. Anal.
- 2010

It is shown that the ADFEM (and the AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

Unstabilized Hybrid High-Order method for a class of degenerate convex minimization problems

- MathematicsSIAM J. Numer. Anal.
- 2021

The application of this HHO method to the class of degenerate convex minimization problems allows for a unique $H(\operatorname{div})$ conforming stress approximation $\sigma_h$ and the main results are a~priori and a posteriori error estimates for the stress error $\s Sigma-\sigma-h$ in Lebesgue norms and a computable lower energy bound.

Convergence of adaptive finite element methods for a nonconvex double-well minimization problem

- MathematicsMath. Comput.
- 2015

This paper focuses on the numerical analysis of a nonconvex variational problem which is related to the relaxation of the two-well problem in the analysis of solid-solid phase transitions with…

The Hybrid High-Order Method for Polytopal Meshes

- Computer Science
- 2020

This monograph provides an introduction to the design and analysis of HHO methods for diffusive problems on general meshes, along with a panel of applications to advanced models in computational mechanics.

Convergence of adaptive FEM for a class of degenerate convex minimization problems

- Mathematics
- 2007

A class of degenerate convex minimization problems allows for some adaptive finite-element method (AFEM) to compute strongly converging stress approximations. The algorithm AFEM consists of…

A BASIC CONVERGENCE RESULT FOR CONFORMING ADAPTIVE FINITE ELEMENTS

- Mathematics
- 2008

We consider the approximate solution with adaptive finite elements of a class of linear boundary value problems, which includes problems of "saddle point" type. For the adaptive algorithm we assume…

Optimality of a Standard Adaptive Finite Element Method

- Mathematics, Computer ScienceFound. Comput. Math.
- 2007

An adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity and does not rely on a recurrent coarsening of the partitions.

A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes

- Mathematics, Computer ScienceMath. Comput.
- 2017

A Hybrid High-Order (HHO) method for steady non-linear Leray–Lions problems is developed and analyzed by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case.