Estimations d'erreur pour des éléments finis droits presque dégénérés

Jamet, Pierre

doi:10.1051/m2an/197610r100431

Cited by 82 publications

(123 citation statements)

References 8 publications

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“…The numerical errors in the H 1 0 and L 2 norms are presented in logarithmic scale respectively in Figures 22 and 23. In Figure 22, we also displayed the curve h/ sin θ m (named "P1 theor") which corresponds to the higher bound (up to a multiplicative constant) of the error estimates obtained by Jamet in [18]. We observe that this curve properly reproduces the behavior of the e1 P 1 (the curves are almost parallel).…”

Section: Numerical Resultsmentioning

confidence: 70%

“…If Hypothesis 5.3 is checked, then the solutionφ is in H 3 (Ω) and we can specify the dependence of the constant C(τ * ), thanks to [18], Theorem 3.1-ii:…”

Section: Moreover Under Assumptions 53 and 56 There Exists A Consmentioning

confidence: 99%

“…These hypotheses explain why this scheme is particularly adapted to very distorted or degenerating grids. In the case of the traditional Lagrange finite elements, it is well-known that a quality criterion of the grid is given by the so-called "maximum angle condition" (see [2,18]): The optimal rate of convergence of the numerical solution toward the solution of the continuous problem, on a family of grids whose step h tends to 0, is obtained if, and only if, the maximum angle of the triangles is bounded away from π, uniformly in h. This property was then extended by Acosta and Durán to the mixed finite elements of Raviart-Thomas type and to the non-conforming finite elements of Crouzeix-Raviart type in [1]. As far as the scheme presented here is concerned, we obtain a sufficient condition of convergence related to the angles of the diamond-cells: each one of these quadrangles can be split along one of its diagonals into two triangles.…”

Section: Introductionmentioning

confidence: 99%

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A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

Domelevo¹,

Omnes²

2005

ESAIM: M2AN

177

251

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Abstract. We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires "Voronoi-type" meshes. We show the equivalence of this finite volume method with a non-conforming finite element method with basis functions being P 1 on the cells, generally called "diamond-cells", of a third mesh. Under geometrical conditions on these diamondcells, we prove a first-order convergence both in the H 1 0 norm and in the L 2 norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergence in the L 2 norm on general grids. They also indicate that this method performs particularly well for the approximation of the gradient of the solution, and may be used on degenerating triangular grids. An example of application on nonconforming locally refined grids is given.Mathematics Subject Classification. 35J05, 35J25, 65N12, 65N15, 65N30.

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Section: Numerical Resultsmentioning

confidence: 70%

“…If Hypothesis 5.3 is checked, then the solutionφ is in H 3 (Ω) and we can specify the dependence of the constant C(τ * ), thanks to [18], Theorem 3.1-ii:…”

Section: Moreover Under Assumptions 53 and 56 There Exists A Consmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

Domelevo¹,

Omnes²

2005

ESAIM: M2AN

177

251

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“…Since the triangulations are nested, we have 91L, C 9lt,+1,/ > 1. The spaces 91L, satisfy the following standard approximation property [4], [5], [16], [21]: If u G Hs, 1 < s < 1 + a, then there exists u, G 91L, such that (2.6) ||m -i*||0 + A,H« -«y||, < C(60, 8" ß)/t/||«||,.…”

mentioning

confidence: 99%

An Optimal Order Process for Solving Finite Element Equations

Bank¹,

Dupont²

1981

Mathematics of Computation

134

239

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Abstract. A ¿-level iterative procedure for solving tbe algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed. The work estimate for this procedure is proportional to the number of unknowns, an optimal order result. General geometry is permitted for the underlying domain, but the shape plays a role in the rate of convergence through elliptic regularity. Finally, a short discussion of the use of this method for parabolic problems is presented.

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“…(We use the term Lagrange element for an approximation that is determined by values of the function being approximated at a finite number of points, in contrast with Hermite elements which also depend on the interpolated function's derivatives.) They showed that instead of requiring that none of the angles in the triangles be small, it suffices to require that none of them be large, i.e., that all angles be bounded away from n. P. Jamet [2] obtained uniform estimates for Lagrange elements under the assumptions that (i) the directions of the sides of the elements are not arbitrarily close to being parallel to any hyperplane and (ii) the space of approximating functions in each element include all polynomials of degree k with k > n/2. The last restriction eliminates linear, bilinear, and trilinear elements in ÍR2 and K3.…”

Section: Introductionmentioning

confidence: 99%

Uniform error estimates for certain narrow Lagrange finite elements

Shenk¹

1994

Math. Comp.

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Abstract. Error estimates of Dupont and Scott are used to derive uniform error estimates for Lagrange finite elements in 5ft" (n > 2) under the following conditions: ( 1 ) The elements can be arbitrarily narrow in any coordinate direction such that a sufficient number of interpolation points are grouped on lines parallel to that coordinate axis, and (2) the space of approximating functions FT in each element T must include the space of polynomials of degree < m -1 for some m > 1 + n/2 . If n is odd, this does not cover elements of lowest degree that are normally considered with the shape regularity requirement that the ratio of their outer and inner diameters be bounded. For example, if n = 3 , the usual requirement with shape regularity is that each Ft contain all first-degree polynomials. The result of this paper requires that each Ft contain all quadratic polynomials, and consequently does not apply to linear (Courant) elements in tetrahedrons or trilinear (tensor) elements in rectangular boxes. Counterexamples in these two cases are included.

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Estimations d'erreur pour des éléments finis droits presque dégénérés

Cited by 82 publications

References 8 publications

A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids

An Optimal Order Process for Solving Finite Element Equations

Uniform error estimates for certain narrow Lagrange finite elements

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