Divergence-free finite elements on tetrahedral grids for 𝑘≥6

Zhang, Shangyou

doi:10.1090/S0025-5718-2010-02412-3

Divergence-free finite elements on tetrahedral grids for $k\ge 6$
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by Shangyou Zhang;

Math. Comp. 80 (2011), 669-695

DOI: https://doi.org/10.1090/S0025-5718-2010-02412-3

Published electronically: August 26, 2010

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Abstract:

It was shown two decades ago that the $P_k$-$P_{k-1}$ mixed element on triangular grids, approximating the velocity by the continuous $P_k$ piecewise polynomials and the pressure by the discontinuous $P_{k-1}$ piecewise polynomials, is stable for all $k\ge 4$, provided the grids are free of a nearly-singular vertex. The problem with the method in 3D was posted then and remains open. The problem is solved partially in this work. It is shown that the $P_k$-$P_{k-1}$ element is stable and of optimal order in approximation, on a family of uniform tetrahedral grids, for all $k\ge 6$. The analysis is to be generalized to non-uniform grids, when we can deal with the complicity of 3D geometry.

For the divergence-free elements, the finite element spaces for the pressure can be avoided in computation, if a classic iterated penalty method is applied. The finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity. Numerical tests are provided.

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.
Daniele Boffi, Three-dimensional finite element methods for the Stokes problem, SIAM J. Numer. Anal. 34 (1997), no. 2, 664–670. MR 1442933, DOI 10.1137/S0036142994270193
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258, DOI 10.1007/978-1-4757-4338-8
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
Michel Fortin and Roland Glowinski, Augmented Lagrangian methods, Studies in Mathematics and its Applications, vol. 15, North-Holland Publishing Co., Amsterdam, 1983. Applications to the numerical solution of boundary value problems; Translated from the French by B. Hunt and D. C. Spicer. MR 724072
J. Pitkäranta and R. Stenberg, Error bounds for the approximation of the Stokes problem using bilinear/constant elements on irregular quadrilateral meshes, The mathematics of finite elements and applications, V (Uxbridge, 1984) Academic Press, London, 1985, pp. 325–334. MR 811045
J. Qin, On the convergence of some low order mixed finite elements for incompressible fluids, Thesis, Pennsylvania State University, 1994.
Jinshui Qin and Shangyou Zhang, Stability and approximability of the $\scr P_1$-$\scr P_0$ element for Stokes equations, Internat. J. Numer. Methods Fluids 54 (2007), no. 5, 497–515. MR 2322456, DOI 10.1002/fld.1407
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
L. R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 1, 111–143 (English, with French summary). MR 813691, DOI 10.1051/m2an/1985190101111
L. R. Scott and M. Vogelius, Conforming finite element methods for incompressible and nearly incompressible continua, Large-scale computations in fluid mechanics, Part 2 (La Jolla, Calif., 1983) Lectures in Appl. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1985, pp. 221–244. MR 818790, DOI 10.1051/m2an/1985190101111
L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
L. R. Scott and S. Zhang, Multilevel Iterated Penalty Method for Mixed Elements, the Proceedings for the Ninth International Conference on Domain Decomposition Methods, 133-139, Bergen, 1998.
Michael Vogelius, A right-inverse for the divergence operator in spaces of piecewise polynomials. Application to the $p$-version of the finite element method, Numer. Math. 41 (1983), no. 1, 19–37. MR 696548, DOI 10.1007/BF01396303
Shangyou Zhang, Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes, Houston J. Math. 21 (1995), no. 3, 541–556. MR 1352605
Shangyou Zhang, A new family of stable mixed finite elements for the 3D Stokes equations, Math. Comp. 74 (2005), no. 250, 543–554. MR 2114637, DOI 10.1090/S0025-5718-04-01711-9
Shangyou Zhang, On the P1 Powell-Sabin divergence-free finite element for the Stokes equations, J. Comput. Math. 26 (2008), no. 3, 456–470. MR 2421893