Abstract
We discuss vertex patch smoothers as overlapping domain decomposition methods for fourth order elliptic partial differential equations. We show that they are numerically very efficient and yield high convergence rates. Furthermore, we discuss low rank tensor approximations for their efficient implementation. Our experiments demonstrate that the inexact local solver yields a method which converges fast and uniformly with respect to mesh refinement and polynomial degree. The multiplicative smoother shows superior performance in terms of solution efficiency, requiring fewer iterations in both two- and three-dimensional cases. Additionally, the solver infrastructure supports a mixed-precision approach, executing the multigrid preconditioner in single precision while performing the outer iteration in double precision, thereby increasing throughput by up to 70 %.
Funding source: China Scholarship Council
Award Identifier / Grant number: 202106380059
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: EXC 2181/1 – 390900948
Funding source: Ministero dell’Università e della Ricerca (MUR)
Award Identifier / Grant number: P2022N5ZNP
Funding source: Istituto Nazionale di Alta Matematica ”Francesco Severi” (INDAM) - Gruppo Nazionale per il Calcolo Scientifico (GNCS)
Award Identifier / Grant number: E53C23001670001
Award Identifier / Grant number: E53C24001950001
Funding statement: Cu Cui was supported by the China Scholarship Council (CSC) under grant no. 202106380059. Francesca Bonizzoni has received support from the project PRIN2022, MUR, Italy, 2023–2025, P2022N5ZNP “SIDDMs: shape-informed data-driven models for parametrized PDEs, with application to computational cardiology”. Francesca Bonizzoni is partially funded by “INdAM – GNCS Project”, codice CUP E53C23001670001. Francesca Bonizzoni is member of INdAM-GNCS. Guido Kanschat was supported by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy EXC 2181/1 – 390900948 (the Heidelberg STRUCTURES Excellence Cluster).
References
[1] D. Arndt, W. Bangerth, M. Bergbauer, M. Feder, M. Fehling, J. Heinz, T. Heister, L. Heltai, M. Kronbichler, M. Maier, P. Munch, J.-P. Pelteret, B. Turcksin, D. Wells and S. Zampini, The deal.II library, version 9.5, J. Numer. Math. 31 (2023), no. 3, 231–246. 10.1515/jnma-2023-0089Search in Google Scholar
[2] D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742–760. 10.1137/0719052Search in Google Scholar
[3]
D. N. Arnold, R. S. Falk and R. Winther,
Preconditioning in
[4]
S. C. Brenner,
[5]
S. C. Brenner and L.-Y. Sung,
[6]
S. C. Brenner and K. Wang,
Two-level additive Schwarz preconditioners for
[7] S. C. Brenner and J. Zhao, Convergence of multigrid algorithms for interior penalty methods, Appl. Numer. Anal. Comput. Math. 2 (2005), no. 1, 3–18. 10.1002/anac.200410019Search in Google Scholar
[8] P. D. Brubeck and P. E. Farrell, A scalable and robust vertex-star relaxation for high-order FEM, SIAM J. Sci. Comput. 44 (2022), no. 5, A2991–A3017. 10.1137/21M1444187Search in Google Scholar
[9] D. Cho, L. F. Pavarino and S. Scacchi, Isogeometric Schwarz preconditioners for the biharmonic problem, Electron. Trans. Numer. Anal. 49 (2018), 81–102. 10.1553/etna_vol49s81Search in Google Scholar
[10] C. Cui, Acceleration of tensor-product operations with tensor cores, ACM Trans. Parallel Comput. 11 (2024), no. 4, Paper No. 15. 10.1145/3695466Search in Google Scholar
[11] C. Cui, P. Grosse-Bley, G. Kanschat and R. Strzodka, An implementation of tensor product patch smoothers on GPUs, SIAM J. Sci. Comput. 47 (2025), no. 2, B280–B307. 10.1137/24M1642706Search in Google Scholar
[12] C. Cui and G. Kanschat, Multigrid methods for the Stokes problem on GPU systems, preprint (2024), https://arxiv.org/abs/2410.09497. 10.2139/ssrn.5009863Search in Google Scholar
[13] C. Cui and G. Kanschat, Multilevel interior penalty methods on GPUs, preprint (2024), https://arxiv.org/abs/2405.18982. Search in Google Scholar
[14] X. Feng and O. A. Karakashian, Two-level non-overlapping Schwarz preconditioners for a discontinuous Galerkin approximation of the biharmonic equation, J. Sci. Comput. 22/23 (2005), 289–314. 10.1007/s10915-004-4141-9Search in Google Scholar
[15] D. Göddeke, R. Strzodka and S. Turek, Performance and accuracy of hardware-oriented native-, emulated- and mixed-precision solvers in FEM simulations, Int. J. Parallel Emergent Distrib. Syst. 22 (2007), no. 4, 221–256. 10.1080/17445760601122076Search in Google Scholar
[16] G. Kanschat, R. Lazarov and Y. Mao, Geometric multigrid for Darcy and Brinkman models of flows in highly heterogeneous porous media: A numerical study, J. Comput. Appl. Math. 310 (2017), 174–185. 10.1016/j.cam.2016.05.016Search in Google Scholar
[17]
G. Kanschat and Y. Mao,
Multigrid methods for
[18]
G. Kanschat and N. Sharma,
Divergence-conforming discontinuous Galerkin methods and
[19] M. Kronbichler and K. Kormann, A generic interface for parallel cell-based finite element operator application, Comput. & Fluids 63 (2012), 135–147. 10.1016/j.compfluid.2012.04.012Search in Google Scholar
[20] M. Kronbichler and K. Kormann, Fast matrix-free evaluation of discontinuous Galerkin finite element operators, ACM Trans. Math. Software 45 (2019), no. 3, Paper No. 29. 10.1145/3325864Search in Google Scholar
[21] R. E. Lynch, J. R. Rice and D. H. Thomas, Direct solution of partial difference equations by tensor product methods, Numer. Math. 6 (1964), 185–199. 10.1007/BF01386067Search in Google Scholar
[22] J. M. Melenk, K. Gerdes and C. Schwab, Fully discrete hp-finite elements: Fast quadrature, Comput. Methods Appl. Mech. Engrg. 190 (2001), no. 32–33, 4339–4364. 10.1016/S0045-7825(00)00322-4Search in Google Scholar
[23] J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hambg. 36 (1971), 9–15. 10.1007/BF02995904Search in Google Scholar
[24] J. Witte, Fast and robust multilevel Schwarz methods using tensor structure for high-order finite elements, PhD thesis, Heidelberg University, 2022. Search in Google Scholar
[25] J. Witte, D. Arndt and G. Kanschat, Fast tensor product Schwarz smoothers for high-order discontinuous Galerkin methods, Comput. Methods Appl. Math. 21 (2021), no. 3, 709–728. 10.1515/cmam-2020-0078Search in Google Scholar
[26]
S. Zhang,
An optimal order multigrid method for biharmonic,
[27] J. Zhao, Convergence of V- and F-cycle multigrid methods for the biharmonic problem using the Hsieh–Clough–Tocher element, Numer. Methods Partial Differential Equations 21 (2005), no. 3, 451–471. 10.1002/num.20048Search in Google Scholar
[28] NVIDIA Corporation, Nsight compute, 2023. Search in Google Scholar
© 2025 Institute of Mathematics of the National Academy of Science of Belarus, published by De Gruyter, Berlin/Boston
Articles in the same Issue
- Frontmatter
- 8th Chinese–German Workshop on Computational and Applied Mathematics
- Contact Problems in Porous Media
- Mapped Coercivity for the Stationary Navier–Stokes Equations and Their Finite Element Approximations
- Super-Localized Orthogonal Decomposition Method for Heterogeneous Linear Elasticity
- A Note on the Quasi-Best Approximation Constant
- Guaranteed Upper Bounds for Iteration Errors and Modified Kačanov Schemes via Discrete Duality
- A Mixed Finite Element Method for Coupled Plates
- Asymptotic Preserving Semi-Implicit Scheme for the Shallow Water Equations with Non-Flat Bottom Topography and Manning Friction Term
- Computing Both Upper and Lower Eigenvalue Bounds by HDG Methods
- High Order Energy Stable Local Discontinuous Galerkin Methods for Camassa–Holm–Novikov Equations
- Tensor-Product Vertex Patch Smoothers for Biharmonic Problems
- High-Order Accurate Structure-Preserving Finite Volume Scheme for Ten-Moment Gaussian Closure Equations with Source Terms: Positivity and Well-Balancedness
- A Staggered Discontinuous Galerkin Method for the Simulation of Wave Propagation in Poroelastic Media
Articles in the same Issue
- Frontmatter
- 8th Chinese–German Workshop on Computational and Applied Mathematics
- Contact Problems in Porous Media
- Mapped Coercivity for the Stationary Navier–Stokes Equations and Their Finite Element Approximations
- Super-Localized Orthogonal Decomposition Method for Heterogeneous Linear Elasticity
- A Note on the Quasi-Best Approximation Constant
- Guaranteed Upper Bounds for Iteration Errors and Modified Kačanov Schemes via Discrete Duality
- A Mixed Finite Element Method for Coupled Plates
- Asymptotic Preserving Semi-Implicit Scheme for the Shallow Water Equations with Non-Flat Bottom Topography and Manning Friction Term
- Computing Both Upper and Lower Eigenvalue Bounds by HDG Methods
- High Order Energy Stable Local Discontinuous Galerkin Methods for Camassa–Holm–Novikov Equations
- Tensor-Product Vertex Patch Smoothers for Biharmonic Problems
- High-Order Accurate Structure-Preserving Finite Volume Scheme for Ten-Moment Gaussian Closure Equations with Source Terms: Positivity and Well-Balancedness
- A Staggered Discontinuous Galerkin Method for the Simulation of Wave Propagation in Poroelastic Media