# Constrained and unconstrained stable discrete minimizations for p-robust local reconstructions in vertex patches in the de Rham complex

@article{ChaumontFrelet2022ConstrainedAU,
title={Constrained and unconstrained stable discrete minimizations for p-robust local reconstructions in vertex patches in the de Rham complex},
author={Th{\'e}ophile Chaumont-Frelet and Martin Vohral{\'i}k},
journal={ArXiv},
year={2022},
volume={abs/2208.05870}
}
• Published 11 August 2022
• Mathematics, Computer Science
• ArXiv
. We analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree p . We show that the discrete minimizers in the spaces of piecewise polynomials of degree p conforming in the H 1 , H ( curl ), or H (div) spaces are as good as the minimizers in these entire (inﬁnite-dimensional) Sobolev spaces, up to a constant that is independent of p . These results are useful in the analysis and design of…
2 Citations

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## References

SHOWING 1-10 OF 37 REFERENCES

### Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem

• Mathematics, Computer Science
Math. Comput.
• 2022
Stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizer over the broken patchwise equilibration is shown.

### Polynomial-degree-robust H(curl)-stability of discrete minimization in a tetrahedron

• Mathematics, Computer Science
ArXiv
• 2020
We prove that the minimizer in the Nedelec polynomial space of some degree p ≥ 0 of a discrete minimization problem performs as well as the continuous minimizer in H(curl), up to a constant that is

### Stable broken H1 and H(div) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions

• Mathematics
Math. Comput.
• 2020
Stability is shown in the sense that the minimizer over piecewise polynomial spaces of the same degree as the data are subordinate in the broken energy norm to the minimizers over the whole broken H and H(div) spaces.

### Equivalence of local-best and global-best approximations in H(curl)

• Mathematics
Calcolo
• 2021
We derive results on equivalence of piecewise polynomial approximations of a given function in the Sobolev space H(curl). We namely show that the global-best approximation of a given H(curl) function

### Equivalence of local-and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div)

• Mathematics, Computer Science
IMA Journal of Numerical Analysis
• 2019
The findings from this work are applied to derive optimal a priori  $hp$-error estimates for mixed and least-squares finite element methods applied to a model diffusion problem.

### Guaranteed, Locally Space-Time Efficient, and Polynomial-Degree Robust a Posteriori Error Estimates for High-Order Discretizations of Parabolic Problems

• Mathematics, Computer Science
SIAM J. Numer. Anal.
• 2017
It is shown that this norm, which is key for local space-time efficiency, is globally equivalent to the $L^2(H^1)\cap H^1(H^{-1})$-norm of the error, with polynomial-degree robust constants.

### On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation

• Computer Science, Mathematics
Numerische Mathematik
• 2021
A novel a posteriori error estimator for conforming finite element discretizations of two- and three-dimensional Helmholtz problems based on an equilibrated flux that is computed by solving patchwise mixed finite element problems is proposed.

### Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction–diffusion problems

• Mathematics
• 2018
We consider energy norm a posteriori error analysis of conforming finite element approximations of singularly perturbed reaction-diffusion problems on simplicial meshes in arbitrary space dimension.

### Polynomial-Degree-Robust A Posteriori Estimates in a Unified Setting for Conforming, Nonconforming, Discontinuous Galerkin, and Mixed Discretizations

• Computer Science, Mathematics
SIAM J. Numer. Anal.
• 2015
Estimates of flux a posteriori error estimates for conforming, nonconforming, discontinuous Galerkin, and mixed finite element discretizations of the two-dimensional Poisson problem are guaranteed, locally computable, locally efficient, and robust with respect to polynomial degree.

### Polynomial extension operators. Part III

• Mathematics
Math. Comput.
• 2012
The existence of a polynomial extension operator in the Sobolev space H(div) is proven constructively and the main application is the existence of commuting projectors with good hp-approximation properties.