The lowest degree of polynomial for a finite element to solve a 2kth-order elliptic equation is k. The Morley element is such a finite element, of polynomial degree 2, for solving a fourth-order biharmonic equation. We design a cubic H3-nonconforming macro-element on two-dimensional triangular grids, solving a sixth-order tri-harmonic equation. We also write down explicitly the 12 basis functions on each macro-element. A convergence theory is established and verified by numerical tests.
Jun Hu, Shangyou Zhang
. A Cubic H3-Nonconforming Finite Element[J]. Communications on Applied Mathematics and Computation, 2019
, 1(1)
: 81
-100
.
DOI: 10.1007/s42967-019-0009-8
1. Alfeld, P., Sirvent, M.:The structure of multivariate superspline spaces of high degree. Math. Comput. 57, 299-308(1991)
2. Argyris, J.H., Fried, I., Scharpf, D.W.:The TUBA family of plate elements for the matrix displacement method. Aeronaut. J. R. Aeronaut. Soc. 72, 514-517(1968)
3. Brenner, S.C., Scott, L.R.:The Mathematical Theory of Finite Element Methods Texts in Applied Mathematics, vol. 15, 3rd edn. Springer, New York (2008)
4. Ciarlet, P.G.:The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
5. Gudi, T.:A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comput. 79, 2169-2189(2010)
6. Hu, J., Ma, R., Shi, Z.:A new a priori error estimate of nonconforming finite element methods. Sci. China Math. 57, 887-902(2014)
7. Hu, J., Huang, Y., Zhang, S.:The lowest order differentiable finite element on rectangular grids. SIAM J. Numer. Anal. 49, 1350-1368(2011)
8. Hu, J., Zhang, S.:The minimal conforming Hk finite element spaces on Rn rectangular grids. Math. Comput. 84, 563-579(2015)
9. Hu, J., Zhang, S.:A canonical construction of Hm-nonconforming triangular finite elements. Ann. Appl. Math. 33, 266-288(2017)
10. Mao, S.P., Shi, Z.C.:On the error bounds of nonconforming finite elements. Sci. China Math. 53, 2917-2926(2010)
11. Powell, M.J.D., Sabin, M.A.:Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3-4, 316-325(1977)
12. Scott, L.R., Zhang, S.:Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483-493(1990)
13. Shi, Z., Wang, M.:Mathematical Theory of Some Nonstandard Finite Element Methods. Computational Mathematics in China, Contemp. Math., vol. 163, pp. 111-125. American Mathematical Society, Providence, RI (1994)
14. Wang, M., Xu, J.:Minimal finite element spaces for 2m-th-order partial differential equations in Rn. Math. Comput. 82, 25-43(2013)
15. Wu, S., Xu, J.:Nonconforming finite element spaces for 2m-th order partial differential equations on Rn simplicial grids when m=n +1. Math. Comp. 88, 531-551(2019)
16. Zhang, S.:A family of 3D continuously differentiable finite elements on tetrahedral grids. Appl. Numer. Math. 59, 219-233(2009)
