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A C1-P2 finite element without nodal basis

Published online by Cambridge University Press:  27 March 2008

Shangyou Zhang
Shangyou Zhang*
Affiliation:
Department of Mathematical Sciences, University of Delaware, DE 19716, USA. [email protected]
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Abstract


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.


Keywords

Differentiable finite elementquadratic elementbiharmonic equationStrang's conjecturecriss-cross gridaveraging interpolationnon-derivative basis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 2 , March 2008 , pp. 175 - 192
Copyright
© EDP Sciences, SMAI, 2008

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References

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).
Billera, L.J., Homology of smooth splines: generic triangulations and a conjecture of Strang. Trans. AMS 310 (1988) 325–340. CrossRef
Bramble, J.H. and Zhang, X., Multigrid methods for the biharmonic problem discretized by conforming C1 finite elements on nonnested meshes. Numer. Functional Anal. Opt. 16 (1995) 835–846. CrossRef
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994).
F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer (1991).
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).
P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. R-2 (1975) 77–84.
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Pub. Inc. (1985).
Hangelbroek, T., Nürnberger, G., Rössl, C., Seidel, H.-P. and Zeilfelder, F., Dimension of C 1-splines on type-6 tetrahedral partitions. J. Approx. Theory 131 (2004) 157–184. CrossRef
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.
Lai, M.-J., Scattered data interpolation and approximation using bivariate C1 piecewise cubic polynomials. Comput. Aided Geom. Design 13 (1996) 81–88. CrossRef
Liu, H., Hong, D. and Cao, D.-Q., 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. CrossRef
Morgan, J. and Scott, L.R., A nodal basis for C 1 piecewise polynomials of degree n. Math. Comp. 29 (1975) 736–740.
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).
Nürnberger, G. and Zeilfelder, F., Developments in bivariate spline interpolation. J. Comput. Appl. Math. 121 (2000) 125–152. CrossRef
Nürnberger, G., Rössl, C., Seidel, H.-P. and Zeilfelder, F., Quasi-interpolation by quadratic piecewise polynomials in three variables. Comput. Aided Geom. Design 22 (2005) 221–249. CrossRef
Nürnberger, G., Rayevskaya, V., Schumaker, L.L. and Zeilfelder, F., Local Lagrange interpolation with bivariate splines of arbitrary smoothness. Constr. Approx. 23 (2006) 33–59. CrossRef
Oswald, P., Hierarchical conforming finite element methods for the biharmonic equation. SIAM J. Numer. Anal. 29 (1992) 1610–1625. CrossRef
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.
Powell, M.J.D. and Sabin, M.A., Piecewise quadratic approximations on triangles. ACM Trans. on Math. Software 3 (1977) 316–325. CrossRef
J. Qin On the convergence of some low order mixed finite elements for incompressible fluids. Ph.D. thesis, Pennsylvania State University, USA (1994).
Qin, J. and Zhang, S., Stability and approximability of the P1-P0 element for Stokes equations. Int. J. Numer. Meth. Fluids 54 (2007) 497–515. CrossRef
P.A. Raviart and V. Girault, Finite element methods for Navier-Stokes equations. Springer (1986).
Schumaker, L.L. and Sorokina, T., A trivariate box macroelement. Constr. Approx. 21 (2005) 413–431. CrossRef
Scott, L.R. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. CrossRef
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.
Strang, G., Piecewise polynomials and the finite element method. Bull. AMS 79 (1973) 1128–1137. CrossRef
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.
Wang, M. and Nonconforming, J. Xu tetrahedral finite elements for fourth order elliptic equations. Math. Comp. 76 (2007) 1–18.
Wang, M. and The Morley, J. Xu element for fourth order elliptic equations in any dimensions. Numer. Math. 103 (2006) 155–169. CrossRef
Zhang, S., An optimal order multigrid method for biharmonic C1 finite element equations. Numer. Math. 56 (1989) 613–624. CrossRef
X. Zhang, Personal communication. University of Maryland, USA (1990).
Zhang, X., Multilevel Schwarz methods for the biharmonic Dirichlet problem. SIAM J. Sci. Comput. 15 (1994) 621–644. CrossRef