arXiv subjects

Théophile Chaumont-Frelet

Publications and source records attributed to Théophile Chaumont-Frelet.

22 records · Page 2Linked to original sources

arXiv2022

$p$-robust equilibrated flux reconstruction in ${\boldsymbol H}(\mathrm{curl})$ based on local minimizations. Application to a posteriori analysis of the curl-curl problem

We present a local construction of H(curl)-conforming piecewise polynomials satisfying a prescribed curl constraint. We start from a piecewise polynomial not contained in the H(curl) space but satisfying a suitable orthogonality property. The procedure employs minimizations in vertex patches and the outcome is, up to a generic constant independent of the underlying polynomial degree, as accurate as the best-approximations over the entire local versions of H(curl). This allows to design guaranteed, fully computable, constant-free, and polynomial-degree-robust a posteriori error estimates of Prager-Synge type for Nédélec finite element approximations of the curl-curl problem. A divergence-free decomposition of a divergence-free H(div)-conforming piecewise polynomial, relying on over-constrained minimizations in Raviart-Thomas spaces, is the key ingredient. Numerical results illustrate the theoretical developments.

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arXiv2021

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

We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show 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 minimizers over the broken H(curl) space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to the a posteriori error analysis of the curl-curl problem discretized with Nédélec finite elements of arbitrary order. The resulting estimators are reliable, locally efficient, polynomial-degree-robust, and inexpensive. They are constructed by a broken patchwise equilibration which, in particular, does not produce a globally H(curl)-conforming flux. The equilibration is only related to edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The error estimates become guaranteed when the regularity pick-up constant is explicitly known. Numerical experiments illustrate the theoretical findings.

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arXiv2020

A generalized finite element method for problems with sign-changing coefficients

Problems with sign-changing coefficients occur, for instance, in the study of transmission problems with metamaterials. In this work, we present and analyze a generalized finite element method in the spirit of the Localized Orthogonal Decomposition, that is especially efficient when the negative and positive materials exhibit multiscale features. We derive optimal linear convergence in the energy norm independently of the potentially low regularity of the exact solution. Numerical experiments illustrate the theoretical convergence rates and show the applicability of the method for a large class of sign-changing diffusion problems.

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arXiv2020

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

We prove that the minimizer in the Nédélec 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 independent of the polynomial degree p. The minimization problems are posed for fields defined on a single non-degenerate tetrahedron in R^3 with polynomial constraints enforced on the curl of the field and its tangential trace on some faces of the tetrahedron. This result builds upon [L. Demkowicz, J. Gopalakrishnan, J. Schöberl SIAM J. Numer. Anal. 47 (2009), 3293--3324] and [M. Costabel, A. McIntosh, Math. Z. 265 (2010), 297--320] and is a fundamental ingredient to build polynomial-degree-robust a posteriori error estimators when approximating the Maxwell equations in several regimes leading to a curl-curl problem.

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