Abstract
Given a tetrahedral grid in 3D, a Powell–Sabin grid can be constructed by refining each original tetrahedron into 12 subtetrahedra. A new divergence-free finite element on 3D Powell–Sabin grids is constructed for Stokes equations, where the velocity is approximated by continuous piecewise quadratic polynomials while the pressure is approximated by discontinuous piecewise linear polynomials on the same grid. To be precise, the finite element space for the pressure is exactly the divergence of the corresponding space for the velocity. Therefore, the resulting finite element solution for the velocity is pointwise divergence-free, including the inter-element boundary. By establishing the inf-sup condition, the finite element is stable and of the optimal order. Numerical tests are provided.
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References
Arnold, D.N., Qin, J.: Quadratic velocity/linear pressure Stokes elements. In: Vichnevetsky, R., Steplemen, R.S. (eds.) Advances in Computer Methods for Partial Differential Equations VII (1992)
Boffi, D.: Three-dimensional finite element methods for the Stokes problem. SIAM J. Numer. Anal. 34, 664–670 (1997)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Fortin, M., Glowinski, R.: Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-value Problems. North Holland, Amsterdam (1983)
Powell, M.J.D., Sabin, M.A.: Piecewise quadratic approximations on triangles. ACM Trans. Math. Softw. 3–4, 316–325 (1977)
Qin, J.: On the convergence of some low order mixed finite elements for incompressible fluids. Thesis, Pennsylvania State University, 1994
Raviart, P.A., Girault, V.: Finite Element Methods for Navier–Stokes Equations. Springer, New York (1986)
Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. Modél. Math. Anal. Numér. 19, 111–143 (1985)
Scott, L.R., Vogelius, M.: Conforming finite element methods for incompressible and nearly incompressible continua. In: Lectures in Applied Mathematics, vol. 22, pp. 221–244 (1985)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)
Scott, L.R., Zhang, S.: Multilevel Iterated Penalty Method for Mixed Elements. In: Proceedings for the Ninth International Conference on Domain Decomposition Methods, pp. 133–139. Bergen (1998)
Stenberg, R.: Analysis of mixed finite element methods for the Stokes problem: a unified approach. Math. Comput. 42, 9–23 (1984)
Zhang, S.: Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes. Houst. J. Math. 21, 541–556 (1995)
Zhang, S.: A new family of stable mixed finite elements for 3D Stokes equations. Math. Comput. 74(240), 543–554 (2005)
Zhang, S.: On the P1 Powell–Sabin divergence-free finite element for the Stokes equations. J. Comput. Math. 26, 456–470 (2008)
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Zhang, S. Quadratic divergence-free finite elements on Powell–Sabin tetrahedral grids. Calcolo 48, 211–244 (2011). https://doi.org/10.1007/s10092-010-0035-4
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DOI: https://doi.org/10.1007/s10092-010-0035-4