×

zbMATH — the first resource for mathematics

A C1-P2 finite element without nodal basis. (English) Zbl 1145.65102
The author introduces a new finite element, which is continuously differentiable, but consists only of piecewise quadratic polynomials on a type of uniform triangulations and constructs a local basis which does not involve nodal values nor derivatives. Different from the traditional finite elements, the author has to construct a special averaging operator which is stable and preserves quadratic polynomials. He obtains the optimal order of approximation of the finite element in interpolation, and in solving the biharmonic equation. Numerical results are provided confirming the analysis.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] D.N. Arnold and J. Qin, Quadratic velocity/linear pressure Stokes elements, in Advances in Computer Methods for Partial Differential EquationsVII, R. Vichnevetsky and R.S. Steplemen Eds. (1992).
[2] L.J. Billera, Homology of smooth splines: generic triangulations and a conjecture of Strang. Trans. AMS310 (1988) 325-340. · Zbl 0718.41017 · doi:10.2307/2001125
[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 0842.65081 · doi:10.1080/01630569508816649
[4] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994). Zbl0811.65093 · Zbl 0811.65093 · eudml:50134
[5] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer (1991). Zbl0788.73002 · Zbl 0788.73002
[6] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). · Zbl 0383.65058
[7] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér.R-2 (1975) 77-84. Zbl0368.65008 · Zbl 0368.65008 · eudml:193271
[8] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Pub. Inc. (1985). · Zbl 0695.35060
[9] T. Hangelbroek, G. Nürnberger, C. Rössl, H.-P. Seidel and F. Zeilfelder, Dimension of C1-splines on type-6 tetrahedral partitions. J. Approx. Theory131 (2004) 157-184. · Zbl 1062.65014 · doi:10.1016/j.jat.2004.09.002
[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 0424.41020
[11] M.-J. Lai, Scattered data interpolation and approximation using bivariate C1 piecewise cubic polynomials. Comput. Aided Geom. Design13 (1996) 81-88. Zbl0873.65011 · Zbl 0873.65011 · doi:10.1016/0167-8396(95)00007-0
[12] H. Liu, D. Hong and D.-Q. Cao, Bivariate C1 cubic spline space over a nonuniform type-2 triangulation and its subspaces with boundary conditions. Comput. Math. Appl.49 (2005) 1853-1865. Zbl1085.41005 · Zbl 1085.41005 · doi:10.1016/j.camwa.2004.08.014
[13] J. Morgan and L.R. Scott, A nodal basis for C1 piecewise polynomials of degree n. Math. Comp.29 (1975) 736-740. Zbl0307.65074 · Zbl 0307.65074 · doi:10.2307/2005284
[14] J. Morgan and L.R. Scott, The dimension of the space of C1 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 0960.41006 · doi:10.1016/S0377-0427(00)00346-0
[16] G. Nürnberger, C. Rössl, H.-P. Seidel and F. Zeilfelder, Quasi-interpolation by quadratic piecewise polynomials in three variables. Comput. Aided Geom. Design22 (2005) 221-249. · Zbl 1082.65009 · doi:10.1016/j.cagd.2004.11.002
[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. Zbl1088.41010 · Zbl 1088.41010 · doi:10.1007/s00365-005-0600-2
[18] P. Oswald, Hierarchical conforming finite element methods for the biharmonic equation. SIAM J. Numer. Anal.29 (1992) 1610-1625. Zbl0771.65071 · Zbl 0771.65071 · doi:10.1137/0729093
[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.
[20] M.J.D. Powell and M.A. Sabin, Piecewise quadratic approximations on triangles. ACM Trans. on Math. Software3 (1977) 316-325. · Zbl 0375.41010 · doi:10.1145/355759.355761
[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. Fluids54 (2007) 497-515. · Zbl 1204.76020 · doi:10.1002/fld.1407
[23] P.A. Raviart and V. Girault, Finite element methods for Navier-Stokes equations. Springer (1986). · Zbl 0585.65077
[24] L.L. Schumaker and T. Sorokina, A trivariate box macroelement. Constr. Approx.21 (2005) 413-431. · Zbl 1077.41009 · doi:10.1007/s00365-004-0572-7
[25] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483-493. · Zbl 0696.65007 · doi:10.2307/2008497
[26] T. Sorokina and F. Zeilfelder, Optimal quasi-interpolation by quadratic C1-splines on type-2 triangulations, in Approximation TheoryXI: Gatlinburg 2004, C.K. Chui, M. Neamtu and L.L. Schumaker Eds., Nashboro Press, Brentwood, TN (2004) 423-438. · Zbl 1074.65015
[27] G. Strang, Piecewise polynomials and the finite element method. Bull. AMS79 (1973) 1128-1137. · Zbl 0285.41009 · doi:10.1090/S0002-9904-1973-13351-8
[28] G. Strang, The dimension of piecewise polynomials, and one-sided approximation, in Conf. on Numerical Solution of Differential Equations, Lecture Notes in Mathematics363, G.A. Watson Ed., Springer-Verlag, Berlin (1974) 144-152. Zbl0279.65091 · Zbl 0279.65091
[29] M. Wang and J. Xu, Nonconforming tetrahedral finite elements for fourth order elliptic equations. Math. Comp.76 (2007) 1-18. · Zbl 1125.65105 · doi:10.1090/S0025-5718-06-01889-8
[30] M. Wang and J. Xu, The Morley element for fourth order elliptic equations in any dimensions. Numer. Math.103 (2006) 155-169. · Zbl 1092.65103 · doi:10.1007/s00211-005-0662-x
[31] S. Zhang, An optimal order multigrid method for biharmonic C1 finite element equations. Numer. Math.56 (1989) 613-624. · Zbl 0667.65089 · doi:10.1007/BF01396347 · eudml:133415
[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 0803.65118 · doi:10.1137/0915041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.