Local finite element approximation of Sobolev differential forms

Gawlik, Evan S.; Holst, Michael; Licht, Martin W.

doi:10.1051/m2an/2021034

Cited by 6 publications

(9 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In fact, this is a common assumption in finite element methods and can easily be implemented in algorithms [20]. The following theorem (see [22,Theorem 5.2, Theorem 5.4]) formalizes the idea and also establishes some important inverse estimates. Theorem 3.3.…”

Section: Rough Degrees Of Freedommentioning

confidence: 98%

“…But only recently have these interpolants been generalized to the vector-valued finite elements that are known as Brezzi-Douglas-Marini, Nédélec, and Raviart-Thomas elements. Several classical and recent results on finite element interpolants have been transferred to the vectorvalued setting over the last years [19,22,7,17].…”

Section: Introductionmentioning

confidence: 99%

“…Generalizing interpolants and their error estimates to vector-valued finite element spaces needs to accommodate that new quality. Successive research [22] has contributed Clément and Scott-Zhang interpolants for finite element vector fields. The Clément interpolants are bounded in Lebesgue spaces but do not satisfy a broken Bramble-Hilbert lemma.…”

Section: Introductionmentioning

confidence: 99%

“…We also establish the notation, which generally follows the literature. Much of the content of this section is a summary of the background given in [22]. We let Ω ⊆ R n be a connected, bounded open set throughout the remainder of this article.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Averaging-based local projections in finite element exterior calculus

Licht¹

2023

Preprint

View full text Add to dashboard Cite

We develop projection operators onto finite element differential forms over simplicial meshes. Our projection is locally bounded in Lebesgue and Sobolev-Slobodeckij norms, uniformly with respect to mesh parameters. Moreover, it incorporates homogeneous boundary conditions and satisfies a local broken Bramble-Hilbert estimate. The construction principle includes the Ern-Guermond projection and a modified Clément-type interpolant with the projection property. The latter seems to be a new result even for Lagrange elements. This projection operator immediately enables an equivalence result on local-and global-best approximations. We combine techniques for the Scott-Zhang and Ern-Guermond projections and adopt the framework of finite element exterior calculus. We instantiate the abstract projection for Brezzi-Douglas-Marini, Nédélec, and Raviart-Thomas elements.2000 Mathematics Subject Classification. 65N30. Key words and phrases. broken Bramble-Hilbert lemma, finite element exterior calculus, Ern-Guermond interpolant.

show abstract

Section: Rough Degrees Of Freedommentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Averaging-based local projections in finite element exterior calculus

Licht¹

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…There have been kinds of interpolators to finite element spaces which work for functions with minimal regularity requirements, such as [16,19,23,24,26,31,32,41]. For these interpolators, the regularization, smoothing or averaging techniques are usually used based on macroelements consisting of patches of elements.…”

Section: Andmentioning

confidence: 99%

Partially adjoint discretizations of adjoint operators

Zhang¹

2022

Preprint

View full text Add to dashboard Cite

This paper concerns the discretizations in pair of adjoint operators between Hilbert spaces so that the adjoint properties can be preserved. Due to the finite-dimensional essence of discretized operators, a new framework, theory of partially adjoint operators, is motivated and presented in this paper, so that adjoint properties can be figured out for finite-dimensional operators which can not be non-trivially densely defined in other background spaces. A formal methodology is presented under the framework to construct partially adjoint discretizations by a conforming discretization (CD) and an accompanied-by-conforming discretization (ABCD) for each of the operators. Moreover, the methodology leads to an asymptotic uniformity of an infinite family of finite-dimensional operators. The validities of the theoretical framework and the formal construction of discretizations are illustrated by a systematic family of in-pair discretizations of the adjoint exterior differential operators.The adjoint properties concerned in the paper are the closed range theorem and the strong dualities, whose preservations have not been well studied yet. Quantified versions of the closed range theorem are established for both adjoint operators and partially adjoint discretizations. The notion Poincaré-Alexander-Lefschetz (P-A-L for short) type duality is borrowed for operator theory, and horizontal and vertical P-A-L dualities are figured out for adjoint operators and their analogues are established for partially adjoint discretizations. Particularly by partially adjoint discretizations of exterior differential operators, the Poincaré-Lefschetz duality is preserved as an identity, which was not yet obtained before. The ABCD, a new kind of discretization method, is motivated by and plays a crucial role in the construction of partially adjoint discretizations. Besides, it can be used for the discretization of single operators; for example, in this paper, de Rham complexes that start with the Crouzeix-Raviart finite element spaces are constructed by series of ABCDs, including both the discretized operators and the domain spaces; commutative diagrams with appropriate regularities are constructed thereon. Equivalences are established between primal and dual discretizations for the elliptic source and eigenvalue problems and the primal and dual mixed discretizations of the Hodge Laplacian problems of exterior differential operators based on the partially adjoint discretizations of adjoint operators. This shows how structure-preserving schemes for equations with linear operators can be lead to by partially adjoint discretizations under the new theoretical framework. SHUO ZHANG 1.3. Organization of the paper 7 1.4. Notations and conventions 8 2. Theory of partially adjoint operators 8 2.1. Basics of adjoint operators revisited 8 2.2. Theory of partially adjoint operators 10 2.3. Some technical proofs 19 3. Partially adjoint discretizations of adjoint operators 23 3.1. Partially adjoint discretizations by CD and ABCD 24 3.2. Characteristics of lo...

show abstract

A stable local commuting projector and optimal hp approximation estimates in $$\varvec{H}(\textrm{curl})$$

Chaumont-Frelet,

Vohralík

2024

Numer. Math.

View full text Add to dashboard Cite

Local finite element approximation of Sobolev differential forms

Cited by 6 publications

References 41 publications

Averaging-based local projections in finite element exterior calculus

Averaging-based local projections in finite element exterior calculus

Partially adjoint discretizations of adjoint operators

A stable local commuting projector and optimal hp approximation estimates in $$\varvec{H}(\textrm{curl})$$

Contact Info

Product

Resources

About