Article
| Title: | Fitted norm preconditioners for the Hodge-Laplacian in mixed form (English) |
| Author: | Boon, Wietse M. |
| Author: | Kraus, Johannes |
| Author: | Luber, Tomáš |
| Author: | Lymbery, Maria |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 70 |
| Issue: | 6 |
| Year: | 2025 |
| Pages: | 907-927 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We use the practical framework for abstract perturbed saddle-point problems recently introduced by Hong et al. to analyze the mixed formulation of the Hodge-Laplace problem on a Hilbert complex. We compose two parameter-dependent norms in which the uniform continuity and stability of the problem follow. This not only guarantees the well-posedness of the corresponding variational formulation on the continuous level, but also of related compatible discrete models. We further simplify the obtained norms and, in both cases, arrive at the same norm-equivalent preconditioner that is easily implementable. The efficiency and uniformity of the preconditioner are demonstrated numerically by the fast convergence and uniformly bounded number of preconditioned MinRes iterations required to solve various instances of Hodge-Laplace problems in two and three space dimensions. (English) |
| Keyword: | preconditioning |
| Keyword: | Hodge-Laplacian |
| Keyword: | perturbed saddle-point |
| Keyword: | Hilbert complex |
| MSC: | 35J05 |
| MSC: | 58A14 |
| MSC: | 58J10 |
| MSC: | 65F08 |
| MSC: | 65N22 |
| MSC: | 65N30 |
| DOI: | 10.21136/AM.2025.0185-25 |
| . | |
| Date available: | 2025-12-20T06:29:38Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153228 |
| . | |
| Reference: | [1] Angoshtari, A., Yavari, A.: Hilbert complexes of nonlinear elasticity.Z. Angew. Math. Phys. 67 (2016), Article ID 143, 30 pages. Zbl 1358.35185, MR 3567615, 10.1007/s00033-016-0735-y |
| Reference: | [2] Arnold, D. N.: Finite Element Exterior Calculus.CBMS-NSF Regional Conference Series in Applied Mathematics 93. SIAM, Philadelphia (2018). Zbl 1506.65001, MR 3908678, 10.1137/1.9781611975543 |
| Reference: | [3] Arnold, D. N., Falk, R. S., Gopalakrishnan, J.: Mixed finite element approximation of the vector Laplacian with Dirichlet boundary conditions.Math. Models Methods Appl. Sci. 22 (2012), Article ID 1250024, 26 pages. Zbl 1260.65100, MR 2974162, 10.1142/S0218202512500248 |
| Reference: | [4] Arnold, D. N., Falk, R. S., Winther, R.: Differential complexes and stability of finite element methods. II. The elasticity complex.Compatible Spatial Discretizations The IMA Volumes in Mathematics and Its Applications 142. Springer, New York (2006), 47-67. Zbl 1119.65399, MR 2249345, 10.1007/0-387-38034-5_3 |
| Reference: | [5] Arnold, D. N., Falk, R. S., Winther, R.: Finite element exterior calculus, homological techniques, and applications.Acta Numerica 15 (2006), 1-155. Zbl 1185.65204, MR 2269741, 10.1017/S0962492906210018 |
| Reference: | [6] Babuška, I.: Error-bounds for finite element method.Numer. Math. 16 (1971), 322-333. Zbl 0214.42001, MR 0288971, 10.1007/BF02165003 |
| Reference: | [7] Ballini, E., Boon, W. M., Fumagalli, A., Scotti, A.: PyGeoN: A Python package for Geo-Numerics.Available at https://zenodo.org/records/13902300. 10.5281/zenodo.13902300 |
| Reference: | [8] Veiga, L. Beirão da, Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D., Russo, A.: Basic principles of virtual element methods.Math. Models Methods Appl. Sci. 23 (2013), 199-214. Zbl 1416.65433, MR 2997471, 10.1142/S0218202512500492 |
| Reference: | [9] Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite element Methods and Applications.Springer Series in Computational Mathematics 44. Springer, Berlin (2013). Zbl 1277.65092, MR 3097958, 10.1007/978-3-642-36519-5 |
| Reference: | [10] Bonaldi, F., Pietro, D. A. Di, Droniou, J., Hu, K.: An exterior calculus framework for polytopal methods.(to appear) in J. Eur. Math. Soc. (JEMS). 10.4171/JEMS/1602 |
| Reference: | [11] Boon, W. M., Holmen, D. F., Nordbotten, J. M., Vatne, J. E.: The Hodge-Laplacian on the Čech-de Rham complex governs coupled problems.J. Math. Anal. Appl. 551 (2025), Article ID 129692, 16 pages. Zbl 08064044, MR 4912322, 10.1016/j.jmaa.2025.129692 |
| Reference: | [12] Boon, W. M., Kuchta, M., Mardal, K.-A., RuiBaier, R.: Robust preconditioners for perturbed saddle-point problems and conservative discretizations of Biot's equations utilizing total pressure.SIAM J. Sci. Comput. 43 (2021), B961--B983. Zbl 07379628, MR 4295052, 10.1137/20M1379708 |
| Reference: | [13] Boon, W. M., Nordbotten, J. M., Vatne, J. E.: Functional analysis and exterior calculus on mixed-dimensional geometries.Ann. Mat. Pura Appl. (4) 200 (2021), 757-789. Zbl 1462.58001, MR 4229549, 10.1007/s10231-020-01013-1 |
| Reference: | [14] Braess, D.: Stability of saddle point problems with penalty.RAIRO, Modélisation Math. Anal. Numér. 30 (1996), 731-742. Zbl 0860.65054, MR 1419936, 10.1051/m2an/1996300607311 |
| Reference: | [15] Brüning, J. W., Lesch, M.: Hilbert complexes.J. Funct. Anal. 108 (1992), 88-132. Zbl 0826.46065, MR 1174159, 10.1016/0022-1236(92)90147-b |
| Reference: | [16] Čap, A., Hu, K.: BGG sequences with weak regularity and applications.Found. Comput. Math. 24 (2024), 1145-1184. Zbl 1555.58013, MR 4783639, 10.1007/s10208-023-09608-9 |
| Reference: | [17] Cheng, A. H.-D.: Poroelasticity.Theory and Applications of Transport in Porous Media 27. Springer, Cham (2016). Zbl 1402.74004, MR 3821571, 10.1007/978-3-319-25202-5 |
| Reference: | [18] Pietro, D. A. Di, Droniou, J.: An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency.Found. Comput. Math. 23 (2023), 85-164. Zbl 1512.65263, MR 4546145, 10.1007/s10208-021-09542-8 |
| Reference: | [19] Evans, L. C.: Partial Differential Equations.Graduate Studies in Mathematics 19. AMS, Providence (2010). Zbl 1194.35001, MR 2597943, 10.1090/gsm/019 |
| Reference: | [20] Falk, R. S., Winther, R.: Local bounded cochain projections.Math. Comput. 83 (2014), 2631-2656. Zbl 1300.65085, MR 3246803, 10.1090/S0025-5718-2014-02827-5 |
| Reference: | [21] Geuzaine, C., Remacle, J.-F.: Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities.Int. J. Numer. Methods Eng. 79 (2009), 1309-1331. Zbl 1176.74181, MR 2566786, 10.1002/nme.2579 |
| Reference: | [22] Hiptmair, R., Pauly, D., Schulz, E.: Traces for Hilbert complexes.J. Func. Anal. 284 (2023), Article ID 109905, 50 pages. Zbl 1519.46021, MR 4557804, 10.1016/j.jfa.2023.109905 |
| Reference: | [23] Hong, Q., Kraus, J., Kuchta, M., Lymbery, M., Mardal, K.-A., Rognes, M. E.: Robust approximation of generalized Biot-Brinkman problems.J. Sci. Comput. 93 (2022), Article ID 77, 28 pages. Zbl 1503.65231, MR 4507134, 10.1007/s10915-022-02029-w |
| Reference: | [24] Hong, Q., Kraus, J., Lymbery, M., Philo, F.: A new practical framework for the stability analysis of perturbed saddle-point problems and applications.Math. Comput. 92 (2023), 607-634. Zbl 1504.65203, MR 4524104, 10.1090/mcom/3795 |
| Reference: | [25] Kraus, J., Lederer, P. L., Lymbery, M., Osthues, K., Schöberl, J.: Hybridized discontinuous Galerkin/hybrid mixed methods for a multiple network poroelasticity model with application in biomechanics.SIAM J. Sci. Comput. 45 (2023), B802--B827. Zbl 1550.76156, MR 4669845, 10.1137/22M149764X |
| Reference: | [26] Lee, J. J.: High order approximation of Hodge Laplace problems with local coderivatives on cubical meshes.ESAIM, Math. Model. Numer. Anal. 56 (2022), 867-891. Zbl 1487.65181, MR 4411484, 10.1051/m2an/2022009 |
| Reference: | [27] Mardal, K.-A., Winther, R.: Preconditioning discretizations of systems of partial differential equations.Numer. Linear Algebra Appl. 18 (2011), 1-40. Zbl 1249.65246, MR 2769031, 10.1002/nla.716 |
| Reference: | [28] Monk, P.: Finite Element Methods for Maxwell's Equations.Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2003). Zbl 1024.78009, MR 2059447, 10.1093/acprof:oso/9780198508885.001.0001 |
| Reference: | [29] Nédélec, J.-C.: Mixed finite elements in $\Bbb{R}^3$.Numer. Math. 35 (1980), 315-341. Zbl 0419.65069, MR 0592160, 10.1007/BF01396415 |
| Reference: | [30] Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for 2nd order elliptic problems.Mathematical Aspects of Finite Element Methods Lecture Notes in Mathematics 606. Springer, Berlin (1977), 292-315. Zbl 0362.65089, MR 0483555, 10.1007/BFb0064470 |
| Reference: | [31] Savostianov, A., Tudisco, F., Guglielmi, N.: Cholesky-like preconditioner for Hodge Laplacians via heavy collapsible subcomplex.SIAM J. Matrix Anal. Appl. 45 (2024), 1827-1849. Zbl 1550.65043, MR 4805864, 10.1137/23M1626396 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
