A C1-P2 finite element without nodal basis

Shangyou Zhang

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 2, page 175-192
  • ISSN: 0764-583X

Abstract

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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.


How to cite

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Zhang, Shangyou. "A C1-P2 finite element without nodal basis." ESAIM: Mathematical Modelling and Numerical Analysis 42.2 (2008): 175-192. <http://eudml.org/doc/250367>.

@article{Zhang2008,
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.
},
author = {Zhang, Shangyou},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Differentiable finite element; quadratic element; biharmonic equation; Strang's conjecture; criss-cross grid; averaging interpolation; non-derivative basis.; differentiable finite element; biharmonic equation; criss-cross grid; non-derivative basis; stability; numerical results},
language = {eng},
month = {3},
number = {2},
pages = {175-192},
publisher = {EDP Sciences},
title = {A C1-P2 finite element without nodal basis},
url = {http://eudml.org/doc/250367},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Zhang, Shangyou
TI - A C1-P2 finite element without nodal basis
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/3//
PB - EDP Sciences
VL - 42
IS - 2
SP - 175
EP - 192
AB - 
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.

LA - eng
KW - Differentiable finite element; quadratic element; biharmonic equation; Strang's conjecture; criss-cross grid; averaging interpolation; non-derivative basis.; differentiable finite element; biharmonic equation; criss-cross grid; non-derivative basis; stability; numerical results
UR - http://eudml.org/doc/250367
ER -

References

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  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.  
  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.  
  4. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994).  
  5. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer (1991).  
  6. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978).  
  7. P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér.R-2 (1975) 77–84.  
  8. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman Pub. Inc. (1985).  
  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.  
  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.  
  11. M.-J. Lai, Scattered data interpolation and approximation using bivariate C1 piecewise cubic polynomials. Comput. Aided Geom. Design13 (1996) 81–88.  
  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.  
  13. J. Morgan and L.R. Scott, A nodal basis for C1 piecewise polynomials of degree n. Math. Comp.29 (1975) 736–740.  
  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.  
  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.  
  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.  
  18. P. Oswald, Hierarchical conforming finite element methods for the biharmonic equation. SIAM J. Numer. Anal.29 (1992) 1610–1625.  
  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.  
  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.  
  23. P.A. Raviart and V. Girault, Finite element methods for Navier-Stokes equations. Springer (1986).  
  24. L.L. Schumaker and T. Sorokina, A trivariate box macroelement. Constr. Approx.21 (2005) 413–431.  
  25. L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483–493.  
  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.  
  27. G. Strang, Piecewise polynomials and the finite element method. Bull. AMS79 (1973) 1128–1137.  
  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.  
  29. M. Wang and J. Xu, Nonconforming tetrahedral finite elements for fourth order elliptic equations. Math. Comp.76 (2007) 1–18.  
  30. M. Wang and J. Xu, The Morley element for fourth order elliptic equations in any dimensions. Numer. Math.103 (2006) 155–169.  
  31. S. Zhang, An optimal order multigrid method for biharmonic C1 finite element equations. Numer. Math.56 (1989) 613–624.  
  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.  

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