De Rham diagram for hp finite element spaces

Demkowicz, Leszek; Monk, Peter; Vardapetyan, L.; Rachowicz, Waldemar

doi:10.1016/s0898-1221(00)00062-6

Cited by 121 publications

(81 citation statements)

References 7 publications

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“…By the proper combination of local h-refinement and local p-enrichment the hp-version achieves tremendously faster convergence rates with respect to the To guarantee stability and convergence of H(div)-conforming FE methods mapping properties analogue to (1.2) have to be valid also on the discrete level, see e.g. [11,16,5]. Approximation results for hp-discretization can be obtained in the course of the de Rham diagram, as done in [14].…”

Section: Introductionmentioning

confidence: 99%

Sparsity Optimized High Order Finite Element Functions on Simplices

Beuchler

Pillwein

Schöberl

et al. 2011

Texts &Amp; Monographs in Symbolic Computation

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This paper deals with conforming high-order finite element discretizations of the vectorvalued function space H(div) in 2 and 3 dimensions. A new set of hierarchic basis functions on simplices with the following two main properties is introduced. Provided an affine simplicial triangulation, first, the divergence of the basis functions is L2-orthogonal, and secondly, the L2-inner product of the interior basis functions is sparse with respect to the polynomial order p. The construction relies on a tensor-product based construction with properly weighted Jacobi polynomials as well as an explicit splitting of the higher-order basis functions into solenoidal and non-solenoidal ones. The basis is suited for fast assembling strategies. The general proof of the sparsity result is done by the assistance of computer algebra software tools. Several numerical experiments document the proved sparsity patterns and practically achieved condition numbers for the parameter-dependent div-div problem. Even on curved elements a tremendous improvement in condition numbers is observed. The precomputed mass and stiffness matrix entries in general form are available online.

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Section: Introductionmentioning

confidence: 99%

Sparsity Optimized High Order Finite Element Functions on Simplices

Beuchler

Pillwein

Schöberl

et al. 2011

Texts &Amp; Monographs in Symbolic Computation

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“…It also highlighted the role of differential forms and algebraic topology in the design and analysis of compatible discretizations. The recent work in [2,8,9,10,22,29,30,39,44,47,52,53,58] and the papers in this volume further affirm that these tools are gaining wider acceptance among mathematicians and engineers. For instance, FE methods that have traditionally relied upon nonconstructive variational [6,18] stability criteria 1 now are being derived by topological approaches that reveal physically relevant degrees of freedom and their proper encoding.…”

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confidence: 88%

“…For instance, FE methods that have traditionally relied upon nonconstructive variational [6,18] stability criteria 1 now are being derived by topological approaches that reveal physically relevant degrees of freedom and their proper encoding. Of particular note are the papers by Arnold et al [4,2] which develop stable finite elements for mixed elasticity, and by Hiptmair [29], Demkowicz et al [22] and Arnold et 1 One exception in FEM was the Grid Decomposition Property (GDP), formulated by Fix et al [26], that gives a topological rather than variational stability condition for mixed discretizations of the Kelvin principle derived from the Hodge decomposition. The GDP is essentially equivalent to an inf-sup condition; see Bochev and Gunzburger [7].…”

mentioning

confidence: 99%

“…As an aside, we point out that developments in the FE literature focus primarily on approximation of differential forms by piecewise polynomials of arbitrary degree [1,3,22] and less on the equivalence between the discrete models. Except in the lowest-order case, such spaces include degrees of freedom that are not cochains and result in differential operators that are not purely combinatorial.…”

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confidence: 99%

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Principles of Mimetic Discretizations of Differential Operators

Bochev

Hyman

The IMA Volumes in Mathematics and Its Applications

156

209

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Abstract. Compatible discretizations transform partial differential equations to discrete algebraic problems that mimic fundamental properties of the continuum equations. We provide a common framework for mimetic discretizations using algebraic topology to guide our analysis. The framework and all attendant discrete structures are put together by using two basic mappings between differential forms and cochains. The key concept of the framework is a natural inner product on cochains which induces a combinatorial Hodge theory on the cochain complex. The framework supports mutually consistent operations of differentiation and integration, has a combinatorial Stokes theorem, and preserves the invariants of the De Rham cohomology groups. This allows, among other things, for an elementary calculation of the kernel of the discrete Laplacian. Our framework provides an abstraction that includes examples of compatible finite element, finite volume, and finite difference methods. We describe how these methods result from a choice of the reconstruction operator and explain when they are equivalent. We demonstrate how to apply the framework for compatible discretization for two scalar versions of the Hodge Laplacian.

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“…Notice that, in order to satisfy requirement (R1), vector functions should be mapped from the master element into the mesh as gradients, see [16,12,33] for details.…”

Section: Analysis Of the Discretized Problemmentioning

confidence: 99%

Full-wave analysis of dielectric waveguides at a given frequency

Vardapetyan

Demkowicz

2002

Math. Comp.

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Abstract. New variational formulation to compute propagation constants is proposed. Based on it, vector finite element method is proved to exclude spurious modes provided finite elements possess discrete compactness property. Convergence analysis is conducted in the framework of collectively compact operators. Reported theoretical results apply to a wide class of vector finite elements including two families of Nedelec and their generalization, the hpedge elements. Numerical experiments fully support theoretical estimates for convergence rates.

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De Rham diagram for hp finite element spaces

Cited by 121 publications

References 7 publications

Sparsity Optimized High Order Finite Element Functions on Simplices

Sparsity Optimized High Order Finite Element Functions on Simplices

Principles of Mimetic Discretizations of Differential Operators

Full-wave analysis of dielectric waveguides at a given frequency

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