AN ERROR ANALYSIS METHOD SPP-BEAM AND A CONSTRUCTION GUIDELINE OF NONCONFORMING FINITE ELEMENTS FOR FOURTH ORDER ELLIPTIC PROBLEMS

doi:10.4208/jcm.1811-m2018-0162

Under two hypotheses of nonconforming finite elements of fourth order elliptic problems, we present a side-patchwise projection based error analysis method (SPP-BEAM for short). Such a method is able to avoid both the regularity condition of exact solutions in the classical error analysis method and the complicated bubble function technique in the recent medius error analysis method. In addition, it is universal enough to admit generalizations. Then, we propose a sufficient condition for these hypotheses by imposing a set of in some sense necessary degrees of freedom of the shape function spaces. As an application, we use the theory to design a P₃ second order triangular H₂ non-conforming element by enriching two P₄ bubble functions and, another P₄ second order triangular H₂ nonconforming finite element, and a P₃ second order tetrahedral H₂ non-conforming element by enriching eight P₄ bubble functions, adding some more degrees of freedom.

Key words: Nonconforming finite element, A priori error analysis, Biharmonic equation

CLC Number:

65N30
73C02

[1] P. Alfeld and M. Sirvent, The structure of multivariate superspline spaces of high degree, Math. Comp., 57(1991), 299-308.

[2] J.H. Argyris, I. Fried and D.W. Scharpf, The TUBA family of plate elements for the matrix displacement method, The Aeronautical Journal of the Royal Aeronautical Society, 72(1968), 514-517.

[3] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008.

[4] H.R. Chen and S.C. Chen, C0-nonconforming elements for a fourth-order elliptic problem, (Chinese) Math. Numer. Sin., 35(2013), 21-30.

[5] H.R. Chen, S.C. Chen and Z.H. Qiao, C0-nonconforming tetrahedral and cuboid elements for the three-dimensional fourth order elliptic problem, Numer. Math., 24(2013), 99-119.

[6] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.

[7] B. Gao, S. Zhang and M. Wang, A note on the nonconforming finite elements for elliptic problems, J. Comput. Math., 29(2011), 215-226.

[8] T. Gudi, A new error analysis for discontinuous finite element methods for linear elliptic problems, Math. Comp., 79(2010), 2169-189.

[9] J. Hu, Y. Huang and S. Zhang, The lowest order differentiable finite element on rectangular grids, SIAM J. Numer. Anal., 49(2011), 135-1368.

[10] J. Hu, R. Ma and Z. Shi, A new a priori error estimate of nonconforming finite element methods, Sci. China Math., 57(2014), 887-902.

[11] J. Hu and S. Zhang, The minimal conforming H^k finite element spaces on Rⁿ rectangular grids, Math. Comp., 84(2015), 563-579.

[12] J. Hu and S. Zhang, A canonical construction of H^m-nonconforming triangular elements, Ann. of Appl. Math., 33(2017), 266-288.

[13] S. P. Mao and Z. C. Shi, On the error bounds of nonconforming finite elements, Sci. China Math., 53(2010), 2917-2926.

[14] M.J.D. Powell and M. A. Sabin, Piecewise quadratic approximations on triangles, ACM Transactions on Mathematical Software, 3-4(1977), 316-325.

[15] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54(1990), 483-493.

[16] Z. Shi and M. Wang, Finite Element Methods (Chinese), Science Press, Beijing (2013).

[17] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley and Teubner, 1996.

[18] M. Wang and J. Xu, The Morley element for fourth order elliptic equations in any dimensions, Numer. Math., 103(2006), 155-169.

[19] M. Wang and J. Xu, Minimal finite element spaces for 2m-th-order partial differential equations in Rⁿ, Math. Comp., 82(2013), 25-43.

[20] M. Wang, P. Zu and S. Zhang, High accuracy nonconforming finite elements for fourth order problems, Sci. China Math., 55(2012), 2183-2192.

[21] A. ?enišek, Alexander Polynomial approximation on tetrahedrons in the finite element method, J. Approx. Theory, 7(1973), 334-351.

[22] S. Zhang, A family of 3D continuously differentiable finite elements on tetrahedral grids, Appl. Numer. Math., 59(2009), 219-233.

[23] S. Zhang, A family of differentiable finite elements on simplicial grids in four space dimensions, Math. Numer. Sin., 38(2016), 309-324.

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Abstract

AN ERROR ANALYSIS METHOD SPP-BEAM AND A CONSTRUCTION GUIDELINE OF NONCONFORMING FINITE ELEMENTS FOR FOURTH ORDER ELLIPTIC PROBLEMS

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