\(H(\operatorname{div})\)-conforming finite element tensors with constraints. (English) Zbl 07925748
Summary: A unified construction of \(H(\operatorname{div})\)-conforming finite element tensors, including vector element, symmetric matrix element, traceless matrix element, and, in general, tensors with linear constraints, is developed in this work. It is based on the geometric decomposition of Lagrange elements into bubble functions on each sub-simplex. Each tensor at a sub-simplex is further decomposed into tangential and normal components. The tangential component forms the bubble function space, while the normal component characterizes the trace. Some degrees of freedom can be redistributed to \((n-1)\)-dimensional faces. The developed finite element spaces are \(H(\operatorname{div})\)-conforming and satisfy the discrete inf-sup condition. Intrinsic bases of the constraint tensor space are also established.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 58J10 | Differential complexes |
| 15A69 | Multilinear algebra, tensor calculus |
Keywords:
\(H(\operatorname{div})\)-conforming finite element; differential form; tensors with linear constraints; geometric decomposition; bubble spaceReferences:
| [1] | Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar, Finite element exterior calculus, homological techniques, and applications, Acta Numer, 15, 1-155, 2006, ISSN: 0962-4929,1474-0508 · Zbl 1185.65204 |
| [2] | Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar, Finite element exterior calculus: From Hodge theory to numerical stability, Bull Amer Math Soc, 47, 2, 281-354, 2010 · Zbl 1207.65134 |
| [3] | Arnold, Douglas N., Finite element exterior calculus, CBMS-NSF Regional Conference Series in Applied Mathematics, xi+120, 2018, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA · Zbl 1506.65001 |
| [4] | Chen, Long; Huang, Xuehai, Decoupling of mixed methods based on generalized Helmholtz decompositions, SIAM J Numer Anal, 56, 5, 2796-2825, 2018 · Zbl 1402.58016 |
| [5] | Chen, Long; Huang, Xuehai, Finite element complexes in two dimensions (in Chinese), Sci. Sin. Math., 54, 1-34, 2024 · Zbl 1530.65159 |
| [6] | Arnold, Douglas N.; Hu, Kaibo, Complexes from complexes, Found Comput Math, 21, 6, 1739-1774, 2021 · Zbl 1520.58011 |
| [7] | Johnson, C.; Mercier, B., Some equilibrium finite element methods for two-dimensional elasticity problems, Numer Math, 30, 1, 103-116, 1978, ISSN: 0029-599X,0945-3245 · Zbl 0427.73072 |
| [8] | Arnold, Douglas N.; Douglas, Jim; Gupta, Chaitan P., A family of higher order mixed finite element methods for plane elasticity, Numer Math, 45, 1, 1-22, 1984, ISSN: 0029-599X,0945-3245 · Zbl 0558.73066 |
| [9] | Arnold, Douglas N.; Winther, Ragnar, Mixed finite elements for elasticity, Numer Math, 92, 401-419, 2002 · Zbl 1090.74051 |
| [10] | Quenneville-Belair, Vincent, A new approach to finite element simulations of general relativity, 113, 2015, ProQuest LLC: ProQuest LLC Ann Arbor, MI, Thesis (Ph.D.)-University of Minnesota |
| [11] | Chen Long, Huang Xuehai. Discrete Hessian complexes in three dimensions. In: The virtual element method and its applications. SEMA SIMAI springer ser., vol. 31, Springer, Cham; 2022, p. 93-135. · Zbl 1503.65291 |
| [12] | Hu, Jun; Liang, Yizhou, Conforming discrete Gradgrad-complexes in three dimensions, Math Comp, 90, 330, 1637-1662, 2021 · Zbl 1479.65028 |
| [13] | Hu, Jun; Zhang, Shangyou, Finite element approximations of symmetric tensors on simplicial grids in \(\mathbb{R}^n\): the lower order case, Math Models Methods Appl Sci, 26, 9, 1649-1669, 2016 · Zbl 1348.65164 |
| [14] | Arnold, Douglas N.; Awanou, Gerard; Winther, Ragnar, Finite elements for symmetric tensors in three dimensions, Math Comp, 77, 263, 1229-1251, 2008 · Zbl 1285.74013 |
| [15] | Adams, Scot; Cockburn, Bernardo, A mixed finite element method for elasticity in three dimensions, J Sci Comput, 25, 3, 515-521, 2005 · Zbl 1125.74382 |
| [16] | Hu, Jun; Zhang, ShangYou, A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids, Sci China Math, 58, 2, 297-307, 2015 · Zbl 1308.74148 |
| [17] | Hu, Jun, Finite element approximations of symmetric tensors on simplicial grids in \(\mathbb{R}^n\): the higher order case, J Comput Math, 33, 3, 283-296, 2015 · Zbl 1340.74095 |
| [18] | Falk, Richard S.; Winther, Ragnar, Local bounded cochain projections, Math Comp, 83, 290, 2631-2656, 2014 · Zbl 1300.65085 |
| [19] | Christiansen, Snorre H.; Munthe-Kaas, Hans Z.; Owren, Brynjulf, Topics in structure-preserving discretization, Acta Numer, 20, 1-119, 2011 · Zbl 1233.65087 |
| [20] | Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar, Geometric decompositions and local bases for spaces of finite element differential forms, Comput Methods Appl Mech Engrg, 198, 21-26, 1660-1672, 2009 · Zbl 1227.65091 |
| [21] | Gopalakrishnan, J.; García-Castillo, L. E.; Demkowicz, L. F., Nédélec spaces in affine coordinates, Comput Math Appl, 49, 7-8, 1285-1294, 2005 · Zbl 1078.65103 |
| [22] | Christiansen, Snorre H.; Hu, Jun; Hu, Kaibo, Nodal finite element de Rham complexes, Numer Math, 139, 2, 411-446, 2018, ISSN: 0029-599X, 0945-3245 · Zbl 1397.65256 |
| [23] | Chen, Long; Huang, Xuehai, Geometric decompositions of the simplex lattice and smooth finite elements in arbitrary dimension, 2021 |
| [24] | Chen, Long; Huang, Xuehai, Finite element de Rham and Stokes complexes in three dimensions, Math Comp, 93, 345, 55-110, 2024, ISSN: 0025-5718,1088-6842 · Zbl 1530.65159 |
| [25] | De Siqueira, Denise; Devloo, Philippe R. B.; Gomes, Sônia M., Hierarchical high order finite element approximation spaces for H(div) and H(curl), (Kreiss, Gunilla; Lötstedt, Per; Målqvist, Axel; Neytcheva, Maya, Numerical mathematics and advanced applications 2009, 2010, Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 269-276 · Zbl 1216.65154 |
| [26] | De Siqueira, Denise; Devloo, Phillipe R. B.; Gomes, Sônia M., A new procedure for the construction of hierarchical high order Hdiv and Hcurl finite element spaces, J Comput Appl Math, 240, 204-214, 2013, ISSN: 0377-0427,1879-1778 · Zbl 1255.65209 |
| [27] | Castro, Douglas A.; Devloo, Philippe R. B.; Farias, Agnaldo M.; Gomes, Sônia M.; de Siqueira, Denise; Durán, Omar, Three dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy, Comput Methods Appl Mech Engrg, 306, 479-502, 2016, ISSN: 0045-7825,1879-2138 · Zbl 1436.65174 |
| [28] | Chen, Long; Huang, Xuehai, Finite elements for div- and divdiv-conforming symmetric tensors in arbitrary dimension, SIAM J Numer Anal, 60, 4, 1932-1961, 2022 · Zbl 1496.65213 |
| [29] | Hu, Kaibo; Lin, Ting; Shi, Bowen, Finite elements for symmetric and traceless tensors in three dimensions, 2023 |
| [30] | Chen, Chunyu; Chen, Long; Huang, Xuehai; Wei, Huayi, Geometric decomposition and efficient implementation of high order face and edge elements, Commun Comput Phys, 35, 4, 1045-1072, 2024, ISSN: 1815-2406,1991-7120 · Zbl 1542.65155 |
| [31] | Nicolaides, R. A., On a class of finite elements generated by Lagrange interpolation. II, SIAM J Numer Anal, 10, 182-189, 1973 · Zbl 0244.65007 |
| [32] | Nédélec, J. C., A new family of mixed finite elements in \(\operatorname{R}^3\), Numer Math, 50, 57-81, 1986 · Zbl 0625.65107 |
| [33] | Brezzi, Franco; Douglas, Jim; Durán, Ricardo; Fortin, Michel, Mixed finite elements for second order elliptic problems in three variables, Numer Math, 51, 2, 237-250, 1987 · Zbl 0631.65107 |
| [34] | Brezzi, Franco; Douglas, Jim; Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numer Math, 47, 2, 217-235, 1985 · Zbl 0599.65072 |
| [35] | Stenberg, Rolf, A nonstandard mixed finite element family, Numer Math, 115, 1, 131-139, 2010, ISSN: 0029-599X, 0945-3245 · Zbl 1189.65285 |
| [36] | Costabel, Martin; McIntosh, Alan, On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math Z, 265, 2, 297-320, 2010 · Zbl 1197.35338 |
| [37] | Huang, Xuehai; Zhang, Chao; Zhou, Yaqian; Zhu, Yangxing, New low-order mixed finite element methods for linear elasticity, Adv Comput Math, 50, 2, 2024, Paper No. 17, 31ISSN: 1019-7168,1572-9044 · Zbl 1541.65156 |
| [38] | Licht, Martin W., On basis constructions in finite element exterior calculus, Adv Comput Math, 48, 2, 2022, Paper No. 14, 36ISSN: 1019-7168,1572-9044 · Zbl 1486.65255 |
| [39] | Chen, Long; Hu, Jun; Huang, Xuehai, Fast auxiliary space preconditioners for linear elasticity in mixed form, Math Comp, 87, 312, 1601-1633, 2018 · Zbl 1447.65174 |
| [40] | Christiansen, Snorre H., On the linearization of Regge calculus, Numer Math, 119, 4, 613-640, 2011, ISSN: 0029-599X, 0945-3245 · Zbl 1269.83022 |
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