An efficient construction of divergence-free spaces in the context of exact finite element de Rham sequences. (English) Zbl 1507.65238
Summary: Exact finite element de Rham subcomplexes relate conforming subspaces in \(\mathrm{H}^1(\varOmega)\), \(\mathrm{H}(\mathrm{curl}; \varOmega)\), \(\mathrm{H}(\mathrm{div}; \varOmega)\), and \(\mathrm{L}^2(\varOmega)\) in a simple way by means of differential operators (gradient, curl, and divergence). The characteristics of such strong couplings are crucial for the design of stable and conservative discretizations of mixed formulations for a variety of multiphysics systems. This work explores these aspects for the construction of divergence-free vector shape functions in a robust fashion allowing stable and faster simulations of mixed formulations of incompressible porous media flows. The resulting schemes are verified by means of numerical tests with known smooth solutions and applied to a benchmark problem to confirm the expected theoretical and computational performance results.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
References:
| [1] | Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO: Anal. NumÉr., 2, 129-151 (1974) · Zbl 0338.90047 |
| [2] | Roberts, J. E.; Thomas, J.-M., Mixed and hybrid methods, (Ciarlet, P. G.; Lions, J. L., Handbook of Numerical Analysis (1991), Elsevier Science Publishers), 527-639 · Zbl 0875.65090 |
| [3] | Boffi, D.; Brezzi, F.; Fortin, M., Mixed Finite Element Methods and Applications (2013), Springer: Springer Heidelberg · Zbl 1277.65092 |
| [4] | Boffi, D.; Fernandes, P.; Gastaldi, L.; Perugia, I., Computational models of electromagnetic resonators: analysis of edge element approximation, SIAM J. Numer. Anal., 36, 4, 1264-1290 (1999) · Zbl 1025.78014 |
| [5] | Ledger, P. D.; Zaglmayr, S., \( h p\)-Finite element simulation of three-dimensional eddy current problems on multiply connected domains, Comput. Methods Appl. Mech. Engrg., 199, 49-52, 3386-3401 (2010) · Zbl 1225.78010 |
| [6] | Arnold, D. N., Differential complexes and numerical stability, (Tatsien, L., Proceedings of the International Congress of Mathematicians. Proceedings of the International Congress of Mathematicians, I: Plenary Lectures and Ceremonies (2002), Higher Education Press: Higher Education Press Beijing), 137-157 · Zbl 1023.65113 |
| [7] | Arnold, D. N.; Falk, R. S.; Winther, R., Differential complexes and stability of finite element methods I. The de rham complex, (Arnold, D. N.; Bochev, P. B.; Lehoucq, R. B.; Nicolaides, R. A.; Shashkov, M., Compatible Spatial Discretizations. Compatible Spatial Discretizations, The IMA Volumes in Mathematics and Its Applications (2006), Springer), 23-46 · Zbl 1119.65398 |
| [8] | Demkowicz, L., Polynomial exact sequences and projection-based interpolation with application to Maxwell equations, (Boffi, D.; Gastaldi, L., Mixed Finite Elements, Compatibility Conditions, and Applications. Mixed Finite Elements, Compatibility Conditions, and Applications, Lecture Notes in Mathematics, vol. 1939 (2008), Springer: Springer Berlin, Heidelberg), 101-158 · Zbl 1143.78366 |
| [9] | Boffi, D.; Brezzi, F.; Fortin, M., Finite elements for the Stokes problem, (Boffi, D.; Gastaldi, L., Mixed Finite Elements, Compatibility Conditions, and Applications. Mixed Finite Elements, Compatibility Conditions, and Applications, Lecture Notes in Mathematics, vol. 1939 (2008), Springer: Springer Berlin, Heidelberg), 45-100 · Zbl 1182.76895 |
| [10] | Cai, W.; Wu, J.; Xin, J., Divergence-free H(div)-conforming hierarchical bases for magnetohydrodynamics (MHD), Comm. Math. Stat., 1, 19-35 (2013) · Zbl 1311.76048 |
| [11] | Rodríguez, A. A.; Camaño, J.; Ghiloni, R.; Vali, A., Graphs, spanning trees and divergence-free finite elements in domains of general topology, IMA J. Numer. Anal., 31, 4, 1986-2003 (2017) · Zbl 1433.65274 |
| [12] | Rodríguez, A. A.; Camaño, J.; De Los Santos, E.; Rapetti, F., A graph approach for the construction of high order divergence-free Raviart-Thomas finite elements, Calcolo, 55, 4, 1-28 (2018) · Zbl 1401.65126 |
| [13] | Devloo, P. R.B.; Durán, O.; Gomes, S. M.; Ainsworth, M., High-order composite finite element exact sequences based on tetrahedral-hexahedral-prismatic-pyramidal partitions, Comput. Methods Appl. Mech. Engrg., 355, 952-975 (2019) · Zbl 1441.65096 |
| [14] | Nédélec, J. C., A new family of mixed finite elements in \(\mathbb{R}^3\), Numer. Math., 50, 57-81 (1986) · Zbl 0625.65107 |
| [15] | Nédélec, J. C., Mixed finite elements in \(\mathbb{R}^3\), Numer. Math., 35, 315-341 (1980) · Zbl 0419.65069 |
| [16] | Dubois, F., Discrete vector potential representation of a divergence-free vector field in three-dimensional domains: numerical analysis of a model problem, SIAM J. Num. Anal., 27, 5, 1103-1141 (1990) · Zbl 0717.65086 |
| [17] | Zaglmayr, S., High Order Finite Element Methods for Electromagnetic Field Computation (2006), Johannes Kepler Universität: Johannes Kepler Universität Linz, (dissertation) |
| [18] | Castro, D. A.; Devloo, P. R.B.; Farias, A. M.; Gomes, S. M.; Siqueira, D.; Durán, O., Three dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy, Comput. Methods Appl. Mech. and Engrg., 306, 479-502 (2016) · Zbl 1436.65174 |
| [19] | Ainsworth, M.; Coyle, J., Hierarchic finite element bases on unstructured tetrahedral meshes, Int. J. Numer. Meth. Engnr., 58, 2103-2130 (2003) · Zbl 1042.65088 |
| [20] | Fuentes, F.; Keith, B.; Demkowicz, L.; Nagaraj, S., Orientation embedded high order shape functions for the exact sequence elements of all shapes, Comput. Math. Appl., 70, 353-458 (2015) · Zbl 1443.65326 |
| [21] | Devloo, P. R.B.; Bravo, C. M.A.; Rylo, C. M.A., Systematic and generic construction of shape functions for p-adaptive meshes of multidimensional finite elements, Comput. Methods Appl. Mech. Engrg., 198, 1716-1725 (2009) · Zbl 1227.65112 |
| [22] | Castro, D. A.; Devloo, P. R.B.; Farias, A. M.; Gomes, S. M.; Durán, O., Hierarchical high order finite element bases for H(div) spaces based on curved meshes for two-dimensional regions or manifolds, J. Comput. Appl. Math., 301, 241-258 (2016) · Zbl 1382.65380 |
| [23] | Farias, A. M.; Devloo, P. R.B.; Gomes, S. M.; Durán, O., An object-oriented framework for multiphysics problems combining different approximation spaces, Finite Elem. Anal. Des., 151, 34-49 (2018) |
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