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Two families of \(H(\operatorname{div})\) mixed finite elements on quadrilaterals of minimal dimension. (English) Zbl 1353.65118

Summary: We develop two families of mixed finite elements on quadrilateral meshes for approximating \((\mathbf u,p)\) solving a second order elliptic equation in mixed form. Standard Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) elements are defined on rectangles and extended to quadrilaterals using the Piola transform, which are well-known to lose optimal approximation of \(\nabla\cdot\mathbf u\). Arnold-Boffi-Falk spaces rectify the problem by increasing the dimension of RT, so that approximation is maintained after Piola mapping. Our two families of finite elements are uniformly inf-sup stable, achieve optimal rates of convergence, and have minimal dimension. The elements for \(\mathbf u\) are constructed from vector polynomials defined directly on the quadrilaterals, rather than being transformed from a reference rectangle by the Piola mapping, and then supplemented by two (one for the lowest order) basis functions that are Piola mapped. One family has full \(H(\operatorname{div})\)-approximation (\(\mathbf u\), \(p\), and \(\nabla\cdot\mathbf u\) are approximated to the same order like RT) and the other has reduced \(H(\operatorname{div})\)-approximation (\(p\) and \(\nabla\cdot\mathbf u\) are approximated to one less power like BDM). The two families are identical except for inclusion of a minimal set of vector and scalar polynomials needed for higher order approximation of \(\nabla\cdot\mathbf u\) and \(p\), and thereby we clarify and unify the treatment of finite element approximation between these two classes. The key result is a Helmholtz-like decomposition of vector polynomials, which explains precisely how a divergence is approximated locally. We develop an implementable local basis and present numerical results confirming the theory.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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[1] D. N. Arnold and G. Awanou, {\it Finite element differential forms on cubical meshes}, Math. Comp., 83 (2014), pp. 1551-1570. · Zbl 1297.65142
[2] D. N. Arnold, D. Boffi, and F. Bonizzoni, {\it Finite element differential forms on curvilinear cubic meshes and their approximation properties}, Numer. Math., 129 (2015), pp. 1-20. · Zbl 1308.65193
[3] D. N. Arnold, D. Boffi, and R. S. Falk, {\it Quadrilateral H(div) finite elements}, SIAM. J. Numer. Anal., 42 (2005), pp. 2429-2451. · Zbl 1086.65105
[4] D. N. Arnold and F. Brezzi, {\it Mixed and nonconforming finite element methods: Implementation, postprocessing and error estimates}, RAIRO Modél. Math. Anal. Numér., 19 (1985), pp. 7-32. · Zbl 0567.65078
[5] D. N. Arnold, R. S. Falk, and R. Winther, {\it Finite element exterior calculus: from Hodge theory to numerical stability}, Bull. Amer. Math. Soc. (N.S.), 47 (2010), pp. 281-354. · Zbl 1207.65134
[6] D. N. Arnold, L. R. Scott, and M. Vogelius, {\it Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon}, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 15 (1988), pp. 169-192. · Zbl 0702.35208
[7] I. Babuska, {\it The finite element method with Lagrangian multipliers}, Numer. Math., 20 (1973), pp. 179-192. · Zbl 0258.65108
[8] P. B. Bochev and D. Ridzal, {\it Rehabilitation of the lowest-order Raviart-Thomas element on quadrilateral grids}, SIAM J. Numer. Anal., 47 (2008), pp. 487-507. · Zbl 1190.65169
[9] D. Boffi, F. Brezzi, and M. Fortin, {\it Mixed Finite Element Methods and Applications}, Springer Ser. Comput. Math. 44, Springer, Heidelberg, 2013. · Zbl 1277.65092
[10] D. Boffi, F. Kikuchi, and J. Schöberl, {\it Edge element computation of Maxwell’s eigenvalues on general quadrilateral meshes}, Math. Models Methods Appl. Sci., 16 (2006),\. 265-273. · Zbl 1097.65113
[11] J. H. Bramble and S. R. Hilbert, {\it Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation}, SIAM J. Numer. Anal., 7 (1970), pp. 112-124. · Zbl 0201.07803
[12] S. C. Brenner and L. R. Scott, {\it The Mathematical Theory of Finite Element Methods}, Springer-Verlag, New York, 1994. · Zbl 0804.65101
[13] F. Brezzi, {\it On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers}, RAIRO, 8 (1974), pp. 129-151. · Zbl 0338.90047
[14] F. Brezzi, J. Douglas, Jr., and L. D. Marini, {\it Two families of mixed elements for second order elliptic problems}, Numer. Math., 47 (1985), pp. 217-235. · Zbl 0599.65072
[15] F. Brezzi and M. Fortin, {\it Mixed and Hybrid Finite Element Methods}, Springer-Verlag, New York, 1991. · Zbl 0788.73002
[16] Z. Chen and J. Douglas, Jr., {\it Prismatic mixed finite elements for second order elliptic problems}, Calcolo, 26 (1989), pp. 135-148. · Zbl 0711.65089
[17] P. G. Ciarlet, {\it The Finite Element Method for Elliptic Problems}, North-Holland, Amsterdam, 1978. · Zbl 0383.65058
[18] M. R. Correa and A. F. D. Loula, {\it Unconditionally stable mixed finite element methods for Darcy flow}, Comput. Methods Appl. Mech. Engrg., 197 (2008), pp. 1525-1540. · Zbl 1194.76109
[19] J. Douglas, Jr. and J. E. Roberts, {\it Global estimates for mixed methods for second order elliptic equations}, Math. Comp., 44 (1985), pp. 39-52. · Zbl 0624.65109
[20] T. Dupont and L. R. Scott, {\it Polynomial approximation of functions in Sobolev space}, Math. Comp., 34 (1980), pp. 441-463. · Zbl 0423.65009
[21] R. S. Falk, P. Gatto, and P. Monk, {\it Hexahedral H(div) and H(curl) finite elements}, ESAIM Math. Model. Numer. Anal., 45 (2011), pp. 115-143. · Zbl 1270.65066
[22] V. Girault and P. A. Raviart, {\it Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms}, Springer-Verlag, Berlin, 1986. · Zbl 0585.65077
[23] J. Li, T. Arbogast, and Y. Huang, {\it Mixed methods using standard conforming finite elements}, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 680-692. · Zbl 1229.76052
[24] J. C. Nédélec, {\it Mixed finite elements in \({\bf R}^3\)}, Numer. Math., 35 (1980), pp. 315-341. · Zbl 0419.65069
[25] A. Rand, A. Gillette, and C. Bajaj, {\it Quadratic serendipity finite elements on polygons using generalized barycentric coordinates}, Math. Comp., 83 (2014). · Zbl 1300.65091
[26] R. A. Raviart and J. M. Thomas, {\it A mixed finite element method for 2nd order elliptic problems}, in Mathematical Aspects of Finite Element Methods, I. Galligani and E. Magenes, eds., Lecture Notes in Math. 606, Springer-Verlag, New York, 1977, pp. 292-315. · Zbl 0362.65089
[27] J. E. Roberts and J.-M. Thomas, {\it Mixed and hybrid methods}, in Handbook of Numerical Analysis, P. G. Ciarlet and J. L. Lions, eds., Vol. 2, North-Holland, Amsterdam, 1991, pp. 523-639. · Zbl 0875.65090
[28] J. Shen, {\it Mixed Finite Element Methods: Analysis and Computational Aspects}, Ph.D. thesis, University of Wyoming, 1992.
[29] J. Shen, {\it Mixed Finite Element Methods on Distorted Rectangular Grids}, Tech. report ISC-94-13-MATH, Institute for Scientific Computation, Texas A&M University, College Station, 1994.
[30] J. M. Thomas, {\it Sur l’analyse numerique des methodes d’elements finis hybrides et mixtes}, Ph.D. thesis, Sciences Mathematiques, à l’Universite Pierre et Marie Curie, 1977.
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