Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates

Arnold, Douglas N.; Brezzi, Franco

doi:10.1051/m2an/1985190100071

Cited by 753 publications

(730 citation statements)

References 11 publications

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“…This saturation assumption is motivated by a superconvergence result that holds true in the case of mixed hybridization, stating that the nonconforming extensionû RT 0 of the multiplier λ h does provide a better approximation of the primal variable u than the piecewise constant approximation u h (see [2,12]). It is easy to see that (6.4) gives rise to an upper and a lower bound for the discretization error of the primal variable u in the L 2 -norm:…”

Section: Error Estimator Based On a Superconvergence Resultsmentioning

confidence: 99%

“…Two estimators η (1) and η (2) are said to be equivalent (locally equivalent), if there exist constants 0 < δ ≤ ∆ (0 < δ T ≤ ∆ T , T ∈ T k ), independent of the refinement level, such that…”

Section: Error Estimators For Raviart-thomas Elements 1349mentioning

confidence: 99%

“…In case of the Poisson equation, this result is well established (cf., e.g., [2,12]). In the general case, it can be proved assuming a discrete H 2 -regularity and using some duality techniques (cf., e.g., [12…”

Section: Proof the Second Equation Of The Variational Problem (21) mentioning

confidence: 99%

“…Furthermore, we assume a = (a ij ) 2 i,j=1 to be a symmetric, matrixvalued function with a ij ∈ L ∞ (Ω), 1 ≤ i, j ≤ 2, and b ∈ L ∞ (Ω) satisfying…”

Section: Introductionmentioning

confidence: 99%

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A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

Wohlmuth

Hoppe

1999

Math. Comp.

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Abstract. We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.

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Section: Error Estimator Based On a Superconvergence Resultsmentioning

confidence: 99%

Section: Error Estimators For Raviart-thomas Elements 1349mentioning

confidence: 99%

Section: Proof the Second Equation Of The Variational Problem (21) mentioning

confidence: 99%

“…Furthermore, we assume a = (a ij ) 2 i,j=1 to be a symmetric, matrixvalued function with a ij ∈ L ∞ (Ω), 1 ≤ i, j ≤ 2, and b ∈ L ∞ (Ω) satisfying…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

Wohlmuth

Hoppe

1999

Math. Comp.

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“…It can be shown that this matrix system is equivalent to some nonconforming finite element method for the original problem (1.1). For the interested readers we refer to [1,7,27].…”

Section: Introductionmentioning

confidence: 99%

Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems

2002

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Abstract. We construct and analyze a mixed finite volume method on quadrilateral grids for elliptic problems written as a system of two first order PDEs in the state variable (e.g., pressure) and its flux (e.g., Darcy velocity). An important point is that no staggered grids or covolumes are used to stabilize the system. Only a single primary grid system is adopted, and the degrees of freedom are imposed on the interfaces. The approximate flux is sought in the lowest-order Raviart-Thomas space and the pressure field in the rotated-Q1 nonconforming space. Furthermore, we demonstrate that the present finite volume method can be interpreted as a rotated-Q1 nonconforming finite element method for the pressure with a simple local recovery of flux. Numerical results are presented for a variety of problems which confirm the usefulness and effectiveness of the method.

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Mesh-centered finite differences from nodal finite elements for elliptic problems

Hennart

Valle

1998

Numer. Methods Partial Differential Eq.

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After it is shown that the classical five-point mesh-centered finite difference scheme can be derived from a low-order nodal finite element scheme by using nonstandard quadrature formulae, higher-order block mesh-centered finite difference schemes for second-order elliptic problems are derived from higher-order nodal finite elements with nonstandard quadrature formulae as before, combined to a procedure known as "transverse integration." Numerical experiments with uniform and nonuniform meshes and different types of boundary conditions confirm the theoretical predictions, in discrete as well as continuous norms.

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Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates

Cited by 753 publications

References 11 publications

A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements

Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems

Mesh-centered finite differences from nodal finite elements for elliptic problems

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