Mixed finite elements on sparse grids

Gradinaru, Vasile; Hiptmair, Ralf

doi:10.1007/s002110100382

Cited by 11 publications

(19 citation statements)

References 25 publications

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“…The remaining columns T j , j > 6, are defined using (20) and the Pj-compatibility property. Let R(rPj)' j = 1,2, ",6, indicate the computable vector in JHtD which represents the right hand side of (17) R(rPjfVE …”

Section: Change Of Basis Inmentioning

confidence: 99%

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Mimetic finite difference method for the Stokes problem on polygonal meshes

Veiga¹,

Gyrya

Lipnikov

et al. 2009

Journal of Computational Physics

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Various approaches to extend the finite element methods to non-traditional elements pyramids, polyhedra, etc.) have been developed over the last decade. Building of basis functions for such elements is a challenging task and may require extensive geometry anal ysis. The mimetic finite difference (MFD) method has many similarities with low-order finite element methods. Both methods try to preserve fundamental properties of physical and mathematical models. The essential difference is that the MFD method uses only the surface representation of discrete unknowns to build stiffness and mass matrices. Since no extension inside the mesh element is required, practical implementation of the MFD method is simple for polygonal meshes that may include degenerate and non-convex el ements. In this article, we develop a MFD method for the Stokes problem on arbitrary polygonal meshes. The method is constructed for tensor coefficients, which will allow to apply it to the linear elasticity problem. The numerical experiments show the second-order convergence for the velocity variable and the first-order for the pressure.

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Section: Change Of Basis Inmentioning

confidence: 99%

“…Various approaches to extend the FE methods to non traditional clements (pyramids, polyhedra, etc) have been developed over the last decade (see, e.g. [20,24,30,31]). Building of basis functions for such elements is a challenging task and extensive geometry analysis.…”

mentioning

confidence: 99%

Mimetic finite difference method for the Stokes problem on polygonal meshes

Veiga¹,

Gyrya

Lipnikov

et al. 2009

Journal of Computational Physics

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“…This results in spaces which are polynomials along edges and on faces, but are non-polynomial within the element. These spaces have been used to construct spaces of 1-forms [12]. More recently, piecewise wholly polynomial bases of scalars (0-forms) on pyramids have been developed [6] and it is this approach we employ here.…”

Section: Pyramidal Elementsmentioning

confidence: 99%

“…This necessarily leads to a requirement for prismatic and pyramidal elements. Development of conforming bases on pyramids is rare and most notable in this context is the work of Gradinaru & Hipmair [12] and Graglia et al [13]. Such functions are not wholly polynomial in nature and as a result are not directly amenable to construction via the Koszul operator employed by Arnold et al…”

Section: Introductionmentioning

confidence: 99%

Conforming Hierarchical Basis Functions

Bluck

2012

Commun. comput. phys.

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Abstract.A unified process for the construction of hierarchical conforming bases on a range of element types is proposed based on an ab initio preservation of the underlying cohomology. This process supports not only the most common simplicial element types, as are now well known, but is generalised to squares, hexahedra, prisms and importantly pyramids. Whilst these latter cases have received (to varying degrees) attention in the literature, their foundation is less well developed than for the simplicial case. The generalisation discussed in this paper is effected by recourse to basic ideas from algebraic topology (differential forms, homology, cohomology, etc) and as such extends the fundamental theoretical framework established by the work of Hiptmair [16][17][18] and Arnold et al. [4] for simplices. The process of forming hierarchical bases involves a recursive orthogonalisation and it is shown that the resulting finite element mass, quasi-stiffness and composite matrices exhibit exponential or better growth in condition number.

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“…These are a special class of forms which can be used to construct continuous forms from discrete forms. However, they can exist only on special domains such as simplices, n-dimensional cubes and shapes, that can be constructed from a cube by collapsing some of its edges [19] such as pyramids or prisms. The numerical variables are then the integrals of the forms over the corresponding figures.…”

Section: Implementation Of the Discrete Equationsmentioning

confidence: 99%

Application of discrete differential forms to spherically symmetric systems in general relativity

Richter¹,

Frauendiener²,

Vogel

2006

Class. Quantum Grav.

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Abstract. In this article we describe applications of Discrete Differential Forms in computational GR. In particular we consider the initial value problem in vacuum space-times that are spherically symmetric. The motivation to investigate this method is mainly its manifest coordinate independence. Three numerical schemes are introduced, the results of which are compared with the corresponding analytic solutions. The error of two schemes converges quadratically to zero. For one scheme the errors depend strongly on the initial data.

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Mixed finite elements on sparse grids

Cited by 11 publications

References 25 publications

Mimetic finite difference method for the Stokes problem on polygonal meshes

Mimetic finite difference method for the Stokes problem on polygonal meshes

Conforming Hierarchical Basis Functions

Application of discrete differential forms to spherically symmetric systems in general relativity

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