Conforming and nonconforming harmonic finite elements

References

• Courant R. Variational methods for the solution of problems of equilibrium and vibration. Bull Amer Math Soc. 1943;49:1–23. doi: 10.1090/S0002-9904-1943-07818-4
   Google Scholar
• Turner MJ, Clough RW, Martin HC, et al. Stiffness and deflection analysis of complex structures. J Aero Sci. 1956;23:805–823. doi: 10.2514/8.3664
   Web of Science ®Google Scholar
• Clough RW. The finite element method in plane stress analysis. Proceedings of the 2nd ASCE Conference on Electronic Computation; Pittsburgh, PA; 1960.
   Google Scholar
• Argyris JH, Fried I, Scharpf DW. The TUBA family of plate elements for the matrix displacement method. Aeronaut J Roy Aeronaut Soc. 1968;72:514–517.
   Web of Science ®Google Scholar
• Babuška I, Aziz AK. Survey lectures on the mathematical foundations of the finite element method. With the collaboration of G. Fix and R. B. Kellogg. In: Aziz AK, editor. The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972). New York: Academic Press; 1972. p. 1–359.
   Google Scholar
• Brenner SC, Scott LR. The mathematical theory of finite element methods. 3rd ed. New York: Springer; 2008. (Texts in applied mathematics; vol. 15).
   Google Scholar
• Ciarlet PG. The finite element method for elliptic problems. Amsterdam: North-Holland; 1978.
   Google Scholar
• Hu J, Huang Y, Zhang S. The lowest order differentiable finite element on rectangular grids. SIAM Num Anal. 2011;49(4):1350–1368. doi: 10.1137/100806497
   Google Scholar
• Hu J, Zhang S. The minimal conforming Hk finite element spaces on Rn rectangular grids. Math Comput. 2015;84(292):563–579. doi: 10.1090/S0025-5718-2014-02871-8
   Google Scholar
• Huang Y, Zhang S. Supercloseness of the divergence-free finite element solutions on rectangular grids. Commun Math Stat. 2013;1(2):143–162. doi: 10.1007/s40304-013-0012-8
   Google Scholar
• Powell MJD, Sabin MA. Piecewise quadratic approximations on triangles. ACM Trans Math Softw. 1977;34:316–325. doi: 10.1145/355759.355761
   Google Scholar
• Schumaker LL, Sorokina T, Worsey AJ. A C1 quadratic trivariate macro-element space defined over arbitrary tetrahedral partitions. J Approx Theory. 2009;158(1):126–142. doi: 10.1016/j.jat.2008.04.014
   Web of Science ®Google Scholar
• Wang M, Xu J. The Morley element for fourth order elliptic equations in any dimensions. Numer Math. 2006;103(1):155–169. doi: 10.1007/s00211-005-0662-x
   Web of Science ®Google Scholar
• Wang M, Xu J. Minimal finite element spaces for 2m-th-order partial differential equations in Rn. Math Comput. 2013;82(281):25–43. doi: 10.1090/S0025-5718-2012-02611-1
   Web of Science ®Google Scholar
• Ženišek A. A general theorem on triangular Cm elements. RAIRO Model Math Anal Numer. 1974;22:119–127.
   Google Scholar
• Zhang S. A C1-P2 finite element without nodal basis. M2AN. 2008;42:175–192. doi: 10.1051/m2an:2008002
   Google Scholar
• Zhang S. A family of 3D continuously differentiable finite elements on tetrahedral grids. Appl Numer Math. 2009;59(1):219–233. doi: 10.1016/j.apnum.2008.02.002
   Web of Science ®Google Scholar
• Zhang S. A family of differentiable finite elements on simplicial grids in four space dimensions. (Chinese) Math Numer Sin. 2016;38(3):309–324.
   Google Scholar
• Arnold DN, Winther R. Mixed finite elements for elasticity. Numer Math. 2002;92(3):401–419. doi: 10.1007/s002110100348
   Web of Science ®Google Scholar
• Hu J, Zhang S. Finite element approximations of symmetric tensors on simplicial grids in Rn: the lower order case. Math Models Methods Appl Sci. 2016;26(9):1649–1669. doi: 10.1142/S0218202516500408
   Google Scholar
• Zhang S. A P4 bubble enriched P3 divergence-free finite element on triangular grids. Comput Math Appl. 2017;74(11):2710–2722. doi: 10.1016/j.camwa.2017.06.036
   Web of Science ®Google Scholar
• Bramble JH, Hilbert SR. Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J Numer Anal. 1970;7:113–124. doi: 10.1137/0707006
   Web of Science ®Google Scholar
• Melenk JM. Operator adapted spectral element methods I. Harmonic and generalized harmonic polynomials. Numer Math. 1999;84(1):35–69. doi: 10.1007/s002110050463
   Web of Science ®Google Scholar
• Hiptmair R, Moiola A, Perugia I, Schwab C. Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM. ESAIM Math Model Numer Anal. 2014;48(3):727–752. doi: 10.1051/m2an/2013137
   Web of Science ®Google Scholar
• Wang J, Ye X. A weak Galerkin finite element method for second-order elliptic problems. J Comput Appl Math. 2013;241:103–115. doi: 10.1016/j.cam.2012.10.003
   Web of Science ®Google Scholar
• Wang J, Ye X. A weak Galerkin mixed finite element method for second order elliptic problems. Math Comput. 2014;83(289):2101–2126. doi: 10.1090/S0025-5718-2014-02852-4
   Web of Science ®Google Scholar
• Mu L, Wang J, Ye X, Zhang S. A C0-weak Galerkin finite element method for the biharmonic equation. J Sci Comput. 2014;59(2):473–495. doi: 10.1007/s10915-013-9770-4
   Web of Science ®Google Scholar
• Beirao da Veiga L, Brezzi F, Cangiani A, Manzini G, Marini LD, Russo A. Basic principles of virtual element methods. Math Models Methods Appl Sci. 2013;23(1):199–214. doi: 10.1142/S0218202512500492
   Web of Science ®Google Scholar
• Beirao da Veiga L, Brezzi F, Marini LD, Russo A. The hitchhiker's guide to the virtual element method. Math Models Methods Appl Sci. 2014;24(8):1541–1573. doi: 10.1142/S021820251440003X
   Web of Science ®Google Scholar
• Scott LR, Zhang S. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comput. 1990;54:483–493. doi: 10.1090/S0025-5718-1990-1011446-7
   Web of Science ®Google Scholar
• Lai M-J, Schumaker LL. Spline functions on triangulations. Cambridge: Cambridge University Press; 2007.
   Google Scholar
• Alfeld P, Sorokina T. Linear differential operators on bivariate spline spaces and spline vector fields. BIT Numer Math. 2016;56(1):15–32. doi: 10.1007/s10543-015-0557-x
   Web of Science ®Google Scholar
• Fortin M, Soulie M. A nonconforming piecewise quadratic finite element on triangles. Int J Numer Methods Eng. 1983;19:505–520. doi: 10.1002/nme.1620190405
   Web of Science ®Google Scholar