Lagrange interpolation and finite element superconvergence

Li, Bo

doi:10.1002/num.10078

Cited by 12 publications

(8 citation statements)

References 17 publications

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“…Therefore, we refer to them as Gauß-Lobatto points. In literature they are also named Jacobi points [37] as they are also the zeros of the orthogonal Jacobi-polynomials P (1,1) p of order p. Now we define the operator I N :…”

Section: Chapter 2 Meshes and Numerical Methods Section 23 Polynommentioning

confidence: 99%

SDFEM with non-standard higher-order finite elements for a convection-diffusion problem with characteristic boundary layers

Franz

2011

Bit Numer Math

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Considering a singularly perturbed problem with exponential and characteristic layers, we show convergence for non-standard higher-order finite elements using the streamline diffusion finite element method (SDFEM). Moreover, for the standard higher-order space Q p supercloseness of the numerical solution w.r.t. an interpolation of the exact solution in the streamline diffusion norm of order p + 1/2 is proved.

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Section: Chapter 2 Meshes and Numerical Methods Section 23 Polynommentioning

confidence: 99%

SDFEM with non-standard higher-order finite elements for a convection-diffusion problem with characteristic boundary layers

Franz

2011

Bit Numer Math

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“…We also point out that ||Π h p − p h || 0,Ω might not superconverge in the case of general mixed elements on triangular meshes, since part of finite element basis functions become more localized in higher order methods (cf. [7,20]). We will present superconvergence results for higher order mixed elements on mildly structured meshes in another paper.…”

Section: Superconvergence For Cr Nonconforming Elementsmentioning

confidence: 99%

Global Superconvergence of the Lowest-Order Mixed Finite Element on Mildly Structured Meshes

Li¹

2018

SIAM J. Numer. Anal.

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In this paper, we develop global superconvergence estimates for the lowest order Raviart-Thomas mixed finite element method for second order elliptic equations with general boundary conditions on triangular meshes, where most pairs of adjacent triangles form approximate parallelograms. In particular, we prove the L 2 -distance between the numerical solution and canonical interpolant for the vector variable is of order 1 + ρ, where ρ ∈ (0, 1] is dependent on the mesh structure. By a cheap local postprocessing operator G h , we prove the L 2 -distance between the exact solution and the postprocessed numerical solution for the vector variable is of order 1 + ρ. As a byproduct, we also obtain the superconvergence estimate for Crouzeix-Raviart nonconforming finite elements on triangular meshes of the same type.

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“…However, we have to recall here a surprising result by Bo Li. In [17] he found that for simplicial P k -type elements with k > d > 1 the standard Lagrange interpolant and the finite element solution are not superclose in the H 1 -norm.…”

Section: The Main Theoremmentioning

confidence: 99%