A C1-P2 finite element without nodal basis

Zhang, Shangyou

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 42 (2008) no. 2, pp. 175-192.

Résumé

A new finite element, which is continuously differentiable, but only piecewise quadratic polynomials on a type of uniform triangulations, is introduced. We construct a local basis which does not involve nodal values nor derivatives. Different from the traditional finite elements, we have to construct a special, averaging operator which is stable and preserves quadratic polynomials. We show the optimal order of approximation of the finite element in interpolation, and in solving the biharmonic equation. Numerical results are provided confirming the analysis.

Détail

DOI : 10.1051/m2an:2008002

Classification : 65N30, 73C35
Keywords: differentiable finite element, quadratic element, biharmonic equation, Strang's conjecture, criss-cross grid, averaging interpolation, non-derivative basis

Bibliographie

[1] D.N. Arnold and J. Qin, Quadratic velocity/linear pressure Stokes elements, in Advances in Computer Methods for Partial Differential Equations VII, R. Vichnevetsky and R.S. Steplemen Eds. (1992).

[2] L.J. Billera, Homology of smooth splines: generic triangulations and a conjecture of Strang. Trans. AMS 310 (1988) 325-340. | Zbl | MR

[3] J.H. Bramble and X. Zhang, Multigrid methods for the biharmonic problem discretized by conforming C1 finite elements on nonnested meshes. Numer. Functional Anal. Opt. 16 (1995) 835-846. | Zbl | MR

[4] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994). | Zbl | MR

[5] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer (1991). | Zbl | MR

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

[7] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. R-2 (1975) 77-84. | Zbl | MR | mathdoc-id

[8] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Pub. Inc. (1985). | Zbl | MR

[9] T. Hangelbroek, G. Nürnberger, C. Rössl, H.-P. Seidel and F. Zeilfelder, Dimension of $C^1$-splines on type-6 tetrahedral partitions. J. Approx. Theory 131 (2004) 157-184. | Zbl | MR

[10] G. Heindl, Interpolation and approximation by piecewise quadratic C1-functions of two variables, in Multivariate Approximation Theory, W. Schempp and K. Zeller Eds., Birkhäuser, Basel (1979) 146-161. | Zbl | MR

[11] M.-J. Lai, Scattered data interpolation and approximation using bivariate C1 piecewise cubic polynomials. Comput. Aided Geom. Design 13 (1996) 81-88. | Zbl | MR

[12] H. Liu, D. Hong and D.-Q. Cao, Bivariate $C^1$ cubic spline space over a nonuniform type-2 triangulation and its subspaces with boundary conditions. Comput. Math. Appl. 49 (2005) 1853-1865. | Zbl | MR

[13] J. Morgan and L.R. Scott, A nodal basis for $C^1$ piecewise polynomials of degree $n$. Math. Comp. 29 (1975) 736-740. | Zbl | MR

[14] J. Morgan and L.R. Scott, The dimension of the space of $C^1$ piecewise-polynomials. Research Report UH/MD 78, Dept. Math., Univ. Houston, USA (1990).

[15] G. Nürnberger and F. Zeilfelder, Developments in bivariate spline interpolation. J. Comput. Appl. Math. 121 (2000) 125-152. | Zbl | MR

[16] G. Nürnberger, C. Rössl, H.-P. Seidel and F. Zeilfelder, Quasi-interpolation by quadratic piecewise polynomials in three variables. Comput. Aided Geom. Design 22 (2005) 221-249. | Zbl | MR

[17] G. Nürnberger, V. Rayevskaya, L.L. Schumaker and F. Zeilfelder, Local Lagrange interpolation with bivariate splines of arbitrary smoothness. Constr. Approx. 23 (2006) 33-59. | Zbl | MR

[18] P. Oswald, Hierarchical conforming finite element methods for the biharmonic equation. SIAM J. Numer. Anal. 29 (1992) 1610-1625. | Zbl | MR

[19] M.J.D. Powell, Piecewise quadratic surface fitting for contour plotting, in Software for Numerical Mathematics, D.J. Evans Ed., Academic Press, New York (1976) 253-2271. | MR

[20] M.J.D. Powell and M.A. Sabin, Piecewise quadratic approximations on triangles. ACM Trans. on Math. Software 3 (1977) 316-325. | Zbl | MR

[21] J. Qin On the convergence of some low order mixed finite elements for incompressible fluids. Ph.D. thesis, Pennsylvania State University, USA (1994).

[22] J. Qin and S. Zhang, Stability and approximability of the P1-P0 element for Stokes equations. Int. J. Numer. Meth. Fluids 54 (2007) 497-515. | MR

[23] P.A. Raviart and V. Girault, Finite element methods for Navier-Stokes equations. Springer (1986). | Zbl | MR

[24] L.L. Schumaker and T. Sorokina, A trivariate box macroelement. Constr. Approx. 21 (2005) 413-431. | Zbl | MR

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

[26] T. Sorokina and F. Zeilfelder, Optimal quasi-interpolation by quadratic $C^1$-splines on type-2 triangulations, in Approximation Theory XI: Gatlinburg 2004, C.K. Chui, M. Neamtu and L.L. Schumaker Eds., Nashboro Press, Brentwood, TN (2004) 423-438. | Zbl | MR

[27] G. Strang, Piecewise polynomials and the finite element method. Bull. AMS 79 (1973) 1128-1137. | Zbl | MR

[28] G. Strang, The dimension of piecewise polynomials, and one-sided approximation, in Conf. on Numerical Solution of Differential Equations, Lecture Notes in Mathematics 363, G.A. Watson Ed., Springer-Verlag, Berlin (1974) 144-152. | Zbl | MR

[29] M. Wang and J. Xu, Nonconforming tetrahedral finite elements for fourth order elliptic equations. Math. Comp. 76 (2007) 1-18. | Zbl | MR

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

[31] S. Zhang, An optimal order multigrid method for biharmonic C1 finite element equations. Numer. Math. 56 (1989) 613-624. | Zbl | MR

[32] X. Zhang, Personal communication. University of Maryland, USA (1990).

[33] X. Zhang, Multilevel Schwarz methods for the biharmonic Dirichlet problem. SIAM J. Sci. Comput. 15 (1994) 621-644. | Zbl | MR