On mixed finite element methods for first order elliptic systems

Fix, George J.; Gunzburger, Max; Nicolaides, R. A.

doi:10.1007/bf01396185

Cited by 60 publications

(44 citation statements)

References 5 publications

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“…Similar arguments can be used to show that 26) which implies that D − N = O(h s ). Formula (4.26) also means that the natural operator N is not compatible with the natural inner and wedge product definitions, while (4.24) means that it is compatible with the reduction map R. Exactly the opposite is true for the derived operator D .…”

Section: Discretementioning

confidence: 78%

“…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].…”

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confidence: 99%

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Principles of Mimetic Discretizations of Differential Operators

Bochev

Hyman

Compatible Spatial Discretizations

153

206

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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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Section: Discretementioning

confidence: 78%

mentioning

confidence: 99%

Principles of Mimetic Discretizations of Differential Operators

Bochev

Hyman

Compatible Spatial Discretizations

153

206

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“…In [7), the following choice of V h and Sh was shown to yield stable and optimally accurate approximations, at least for polygonal domains. First, we subdivide 0 into quadrilaterals, and then subdivide each quadrilateral into four triangles by drawing the diagonals.…”

Section: Dual Methods For Second-order Elliptic Equationsmentioning

confidence: 99%

“…For V h we take all continuous piecewise linear vector fields ~ith respect to the resulting triangulation and then define Sh = div V h • The resulting space Sh can be shown to be a subspace of all piecewise constants over the triangulation. See [7] for details.…”

Section: Dual Methods For Second-order Elliptic Equationsmentioning

confidence: 99%

On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations

Gunzburger

Nicolaides

1986

Applied Numerical Mathematics

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“…It was shown in [33] that the GDP, i.e., (31), along with the relation S 3 k = ∇ · (S 2 k ), are necessary and sufficient for the stability of the discretized Kelvin principle (27).…”

Section: The Grid Decomposition Propertymentioning

confidence: 99%

Least Squares Finite Element Methods

Bochev¹,

Gunzburger²

2015

Encyclopedia of Applied and Computational Mathematics

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Abstract. Least-squares finite element methods are an attractive class of methods for the numerical solution of partial differential equations. They are motivated by the desire to recover, in general settings, the advantageous features of Rayleigh-Ritz methods such as the avoidance of discrete compatibility conditions and the production of symmetric and positive definite discrete systems. The methods are based on the minimization of convex functionals that are constructed from equation residuals. This paper focuses on theoretical and practical aspects of least-square finite element methods and includes discussions of what issues enter into their construction, analysis, and performance. It also includes a discussion of some open problems.

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On mixed finite element methods for first order elliptic systems

Cited by 60 publications

References 5 publications

Principles of Mimetic Discretizations of Differential Operators

Principles of Mimetic Discretizations of Differential Operators

On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations

Least Squares Finite Element Methods

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