A family of 3D H2-nonconforming tetrahedral finite elements for the biharmonic equation

Hu, Jun; Tian, Shudan; Zhang, Shangyou

doi:10.1007/s11425-019-1661-8

Cited by 12 publications

(3 citation statements)

References 17 publications

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“…However, to the best of our knowledge, optimal piecewise cubic or quartic finite element schemes (either conforming or nonconforming) for a planar biharmonic equation have not been discovered. We remark that several O(h 2 ) ordered finite element methods are designed with piecewise cubic polynomials enriched with higher-degree bubbles (see, e.g., [24,30,31,49]), though we are concerned with methods with exactly piecewise cubic polynomials. For a biharmonic problem in higher dimensions and other problems with higher orders, greater difficulties can be expected.…”

Section: A Brief Review Of Relevant Workmentioning

confidence: 99%

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

Zhang

2021

Sci. China Math.

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This paper presents a nonconforming finite element scheme for the planar biharmonic equation, which applies piecewise cubic polynomials (P 3 ) and possesses O(h 2 ) convergence rate for smooth solutions in the energy norm on general shape-regular triangulations. Both Dirichlet and Navier type boundary value problems are studied. The basis for the scheme is a piecewise cubic polynomial space, which can approximate the H 4 functions with O(h 2 ) accuracy in the broken H 2 norm. Besides, a discrete strengthened Miranda-Talenti, which is usually not true for nonconforming finite element spaces, is proved. The finite element space does not correspond to a finite element defined with Ciarlet's triple; however, it admits a set of locally supported basis functions and can thus be implemented by the usual routine. The notion of the finite element Stokes complex plays an important role in the analysis as well as the construction of the basis functions.

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Section: A Brief Review Of Relevant Workmentioning

confidence: 99%

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

Zhang

2021

Sci. China Math.

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“…1 in [39], and other higher order nonconforming methods [9,17,20,25,29,40]. The other way is to adopt different variational principles to avoid computational difficulty.…”

Section: Introductionmentioning

confidence: 99%

A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids

Hu¹,

Ma²,

Zhang³

2020

Preprint

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This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric stress σ = −∇ 2 u is sought in the Sobolev space H(divdiv, Ω; S) simultaneously with the displacement u in L 2 (Ω). Stemming from the structure of H(div, Ω; S) conforming elements for the linear elasticity problems proposed by J. Hu and S. Zhang, the H(divdiv, Ω; S) conforming finite element spaces are constructed by imposing the normal continuity of divσ on the H(div, Ω; S) conforming spaces of P k symmetric tensors. The inheritance makes the basis functions easy to compute. The discrete spaces for u are composed of the piecewise P k−2 polynomials without requiring any continuity. Such mixed finite elements are inf-sup stable on both triangular and tetrahedral grids for k ≥ 3, and the optimal order of convergence is achieved. Besides, the superconvergence and the postprocessing results are displayed. Some numerical experiments are provided to demonstrate the theoretical analysis.

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“…As a consequence, the C 0 ‐continuity requirement is no longer needed. One can consult the very recent contributions due to Hu et al [11, 12] over tetrahedrons as a successful example of this type. Another approach is to utilize the special properties of the shape function space and the symmetry of the element shape.…”

Section: Introductionmentioning

confidence: 99%

High accuracy nonconforming biharmonic element over n‐rectangular meshes

Zhou

Meng

2020

Numerical Methods Partial

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This work gives the high accuracy analysis of a rectangular biharmonic element in arbitrarily high-dimensional cases. Given an n-rectangle, we construct the nonconforming finite element and show its explicit standard basis representation. We prove that, if the n-rectangular mesh is uniform, this element can achieve a second order convergence rate in energy norm when applied to biharmonic problems. Numerical examples for n = 3 are also presented.

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A family of 3D H2-nonconforming tetrahedral finite elements for the biharmonic equation

Cited by 12 publications

References 17 publications

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

An optimal piecewise cubic nonconforming finite element scheme for the planar biharmonic equation on general triangulations

A family of mixed finite elements for the biharmonic equations on triangular and tetrahedral grids

High accuracy nonconforming biharmonic element over n‐rectangular meshes

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