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Kaibo Hu1, Ragnar Winther2   

  1. 1. School of Mathematics, University of Minnesota, 55455 Minneapolis, MN, USA;
    2. Department of Mathematics, University of Oslo, 0316 Oslo, Norway
  • Received:2018-04-27 Revised:2018-04-27 Online:2021-05-15 Published:2021-04-12
  • Contact: Kaibo Hu,
  • Supported by:
    The authors are grateful to Douglas N. Arnold, Richard S. Falk and Jinchao Xu for several discussions about the results of this paper. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 339643. The research of the first author leading to the results of this paper was partly carried out during his affiliation with the University of Oslo, partly supported by China Scholarship Council (CSC), project 201506010013.

Kaibo Hu, Ragnar Winther. WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS[J]. Journal of Computational Mathematics, 2021, 39(3): 333-357.

The purpose of this paper is to discuss representations of high order C0 finite element spaces on simplicial meshes in any dimension. When computing with high order piecewise polynomials the conditioning of the basis is likely to be important. The main result of this paper is a construction of representations by frames such that the associated L2 condition number is bounded independently of the polynomial degree. To our knowledge, such a representation has not been presented earlier. The main tools we will use for the construction is the bubble transform, introduced previously in[1], and properties of Jacobi polynomials on simplexes in higher dimensions. We also include a brief discussion of preconditioned iterative methods for the finite element systems in the setting of representations by frames.

CLC Number: 

[1] R.S. Falk, R. Winther, The bubble transform:A new tool for analysis of finite element methods, Foundations of Computational Mathematics, (2013), 1-32.
[2] C. Bernardi, Y. Maday, Spectral methods, Handbook of numerical analysis, 5(1997), 209-485.
[3] N. Hu, X.Z. Guo, I. Katz, Bounds for eigenvalues and condition numbers in the p-version of the finite element method, Mathematics of Computation of the American Mathematical Society, 67:224(1998), 1423-1450.
[4] C. Schwab, p-and hp-finite element methods:Theory and applications in solid and fluid mechanics, Oxford University Press, 1998.
[5] B.A. Szabo, I. Babuška, Finite element analysis, John Wiley & Sons, 1991.
[6] S. Beuchler, V. Pillwein, J. Schöberl, S. Zaglmayr, Sparsity optimized high order finite element functions on simplices, Numerical and Symbolic Scientific Computing, pages 21-44, Springer, 2012.
[7] T. Führer, J.M. Melenk, D. Praetorius, A. Rieder, Optimal additive Schwarz methods for the hp-BEM:The hypersingular integral operator in 3D on locally refined meshes, Computers & Mathematics with Applications, 70:7(2015), 1583-1605.
[8] G. Karniadakis, S. Sherwin, Spectral/hp element methods for computational fluid dynamics, Oxford University Press, 2013.
[9] H. Li, J. Shen, Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle, Mathematics of Computation, 79:271(2010), 1621-1646.
[10] S. Zaglmayr, High order finite element methods for electromagnetic field computation, PhD thesis, J. K. University Linz, 2006.
[11] J. Xin, W. Cai, Well-conditioned orthonormal hierarchical L2 bases on ${\Bbb R}$n simplicial elements, Journal of scientific computing, 50:2(2012), 446-461.
[12] C.F. Dunkl, Y. Xu, Orthogonal polynomials of several variables, Cambridge University Press, 2014.
[13] Z. Ciesielskii, J. Domsta, The degenerate b-spline basis as basis in the space of algebraic polynomials, Ann. Polon. Math, 26(1985), 71-79.
[14] T. Lyche, K. Scherer, On the p-norm condition number of the multivariate triangular Bernstein basis, Journal of computational and applied mathematics, 119:1(2000), 259-273.
[15] P. Oswald, Frames and space splittings in Hilbert spaces, Lectures Notes Part I, Bell. Labs, 1997.
[16] T. Koornwinder, Two-variable analogues of the classical orthogonal polynomials, Theory and applications of special functions, (1975), 435-495.
[17] L. Zhang, T. Cui, H. Liu, A set of symmetric quadrature rules on triangles and tetrahedra, Journal of Computational Mathematics, (2009), 89-96.
[18] K.A. Mardal, R. Winther, Preconditioning discretizations of systems of partial differential equations, Numerical Linear Algebra with Applications, 18:1(2011), 1-40.
[19] J. Xu, Iterative methods by space decomposition and subspace correction, SIAM review, 34:4(1992), 581-613.
[20] J. Xu, L.T. Zikatanov, Algebraic multigrid methods, arXiv preprint arXiv:1611.01917, 2016.
[21] Y.J. Lee, J. Wu, J. Xu, L. Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM journal on matrix analysis and applications, 28:3(2006), 634-641.
[22] J. Schöberl, J.M. Melenk, C. Pechstein, S. Zaglmayr, Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements, IMA Journal of Numerical Analysis, 28:1(2008), 1-24.
[23] J. Xin, W. Cai, A well-conditioned hierarchical basis for triangular H(curl)-conforming elements, Commun. Comput. Phys, 9:3(2011), 780-806.
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