The minimal conforming 𝐻^{𝑘} finite element spaces on 𝑅ⁿ rectangular grids

Hu, Jun; Zhang, Shangyou

doi:10.1090/S0025-5718-2014-02871-8

The minimal conforming $H^k$ finite element spaces on $R^n$ rectangular grids
HTML articles powered by AMS MathViewer

by Jun Hu and Shangyou Zhang PDF

Math. Comp. 84 (2015), 563-579 Request permission

Abstract:

A family of $C^{k-1}$-$Q_{k}$ finite elements on $R^n$ rectangular grids is constructed. The finite element space is shown to be the full $C^{k-1}$-$Q_{k}$ space and possess the optimal order of approximation property. The polynomial degree is minimal in order to form such a $H^{k}$ finite element space. Numerical tests are provided for using the 2D $C^1$-$Q_{2}$ and $C^2$-$Q_{3}$ finite elements.

References

Vilhelm Adolfsson and Jill Pipher, The inhomogeneous Dirichlet problem for $\Delta ^2$ in Lipschitz domains, J. Funct. Anal. 159 (1998), no. 1, 137–190. MR 1654182, DOI 10.1006/jfan.1998.3300
Peter Alfeld and Maritza Sirvent, The structure of multivariate superspline spaces of high degree, Math. Comp. 57 (1991), no. 195, 299–308. MR 1079007, DOI 10.1090/S0025-5718-1991-1079007-2
M. S. Agranovich, On the theory of Dirichlet and Neumann problems for linear strongly elliptic systems with Lipschitz domains, Funktsional. Anal. i Prilozhen. 41 (2007), no. 4, 1–21, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 41 (2007), no. 4, 247–263. MR 2411602, DOI 10.1007/s10688-007-0023-x
J. H. Argyris, I. Fried, D. W. Scharpf, The TUBA family of plate elements for the matrix displacement method, The Aeronautical Journal of the Royal Aeronautical Society 72 (1968), pp. 514–517.
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Monique Dauge, Problèmes de Neumann et de Dirichlet sur un polyèdre dans $\textbf {R}^3$: régularité dans des espaces de Sobolev $L_p$, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 1, 27–32 (French, with English summary). MR 954085
Monique Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR 961439, DOI 10.1007/BFb0086682
V. Girault and L. R. Scott, Hermite interpolation of nonsmooth functions preserving boundary conditions, Math. Comp. 71 (2002), no. 239, 1043–1074. MR 1898745, DOI 10.1090/S0025-5718-02-01446-1
J. Hu, Y. Q. Huang and Q. Lin. The lower bounds for eigenvalues of elliptic operators-by nonconforming finite element methods. arXiv:1112.1145v2[math.NA], 2013.
Jun Hu, Yunqing Huang, and Shangyou Zhang, The lowest order differentiable finite element on rectangular grids, SIAM J. Numer. Anal. 49 (2011), no. 4, 1350–1368. MR 2817542, DOI 10.1137/100806497
Jun Hu and Zhong-ci Shi, Constrained quadrilateral nonconforming rotated ${\scr Q}_1$ element, J. Comput. Math. 23 (2005), no. 6, 561–586. MR 2190317
Jun Hu and Zhong-Ci Shi, The best $L^2$ norm error estimate of lower order finite element methods for the fourth order problem, J. Comput. Math. 30 (2012), no. 5, 449–460. MR 2988473, DOI 10.4208/jcm.1203-m3855
David Jerison and Carlos E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), no. 1, 161–219. MR 1331981, DOI 10.1006/jfan.1995.1067
Vladimir Kozlov and Vladimir Maz′ya, Asymptotic formula for solutions to elliptic equations near the Lipschitz boundary, Ann. Mat. Pura Appl. (4) 184 (2005), no. 2, 185–213. MR 2149092, DOI 10.1007/s10231-004-0108-6
Heejeong Lee and Dongwoo Sheen, A new quadratic nonconforming finite element on rectangles, Numer. Methods Partial Differential Equations 22 (2006), no. 4, 954–970. MR 2230281, DOI 10.1002/num.20131
Chunjae Park and Dongwoo Sheen, $P_1$-nonconforming quadrilateral finite element methods for second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), no. 2, 624–640. MR 2004191, DOI 10.1137/S0036142902404923
M. J. D. Powell and M. A. Sabin, Piecewise quadratic approximations on triangles, ACM Trans. Math. Software 3 (1977), no. 4, 316–325. MR 483304, DOI 10.1145/355759.355761
Larry L. Schumaker, Spline functions: basic theory, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2007. MR 2348176, DOI 10.1017/CBO9780511618994
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
Chung Tze Shih, On a spline finite element method, Math. Numer. Sinica 1 (1979), no. 1, 50–72 (Chinese, with English summary). MR 656879
Ming Wang and Jinchao Xu, Minimal finite element spaces for $2m$-th-order partial differential equations in $R^n$, Math. Comp. 82 (2013), no. 281, 25–43. MR 2983014, DOI 10.1090/S0025-5718-2012-02611-1
Ming Wang, Zhong-Ci Shi, and Jinchao Xu, Some $n$-rectangle nonconforming elements for fourth order elliptic equations, J. Comput. Math. 25 (2007), no. 4, 408–420. MR 2337403
Xuejun Xu and Shangyou Zhang, A new divergence-free interpolation operator with applications to the Darcy-Stokes-Brinkman equations, SIAM J. Sci. Comput. 32 (2010), no. 2, 855–874. MR 2609343, DOI 10.1137/090751049
Shangyou Zhang, A family of 3D continuously differentiable finite elements on tetrahedral grids, Appl. Numer. Math. 59 (2009), no. 1, 219–233. MR 2474112, DOI 10.1016/j.apnum.2008.02.002
Shangyou Zhang, On the full $C_1$-$Q_k$ finite element spaces on rectangles and cuboids, Adv. Appl. Math. Mech. 2 (2010), no. 6, 701–721. MR 2719052, DOI 10.4208/aamm.09-m0993