Abstract
This paper is concerned with the extension of the algebraic flux-correction (AFC) approach (Kuzmin in Computational fluid and solid mechanics, Elsevier, Amsterdam, pp 887–888, 2001; J Comput Phys 219:513–531, 2006; Comput Appl Math 218:79–87, 2008; J Comput Phys 228:2517–2534, 2009; Flux-corrected transport: principles, algorithms, and applications, 2nd edn. Springer, Berlin, pp 145–192, 2012; J Comput Appl Math 236:2317–2337, 2012; Kuzmin et al. in Comput Methods Appl Mech Eng 193:4915–4946, 2004; Int J Numer Methods Fluids 42:265–295, 2003; Kuzmin and Möller in Flux-corrected transport: principles, algorithms, and applications. Springer, Berlin, 2005; Kuzmin and Turek in J Comput Phys 175:525–558, 2002; J Comput Phys 198:131–158, 2004) to nonconforming finite element methods for the linear transport equation. Accurate nonoscillatory approximations to convection-dominated flows are obtained by stabilizing the continuous Galerkin method by solution-dependent artificial diffusion. Its magnitude is controlled by a flux limiter. This concept dates back to flux-corrected transport schemes. The unique feature of AFC is that all information is extracted from the system matrices which are manipulated to satisfy certain mathematical constraints. AFC schemes have been devised with conforming \(P_1\) and \(Q_1\) finite elements in mind but this is not a prerequisite. Here, we consider their extension to the nonconforming Crouzeix–Raviart element (Crouzeix and Raviart in RAIRO R3 7:33–76, 1973) on triangular meshes and its quadrilateral counterpart, the class of rotated bilinear Rannacher–Turek elements (Rannacher and Turek in Numer Methods PDEs 8:97–111, 1992). The underlying design principles of AFC schemes are shown to hold for (some variant of) both elements. However, numerical tests for a purely convective flow and a convection–diffusion problem demonstrate that flux-corrected solutions are overdiffusive for the Crouzeix–Raviart element. Good resolution of smooth and discontinuous profiles is attested to \(Q_1^\mathrm{nc}\) approximations on quadrilateral meshes. A synthetic benchmark is used to quantify the artificial diffusion present in conforming and nonconforming high-resolution schemes of AFC-type. Finally, the implementation of efficient sparse matrix–vector multiplications is addressed.
Similar content being viewed by others
References
Anderson DG (1965) Iterative procedure for nonlinear integral equations. J Assoc Comput Mach 12:547–560
Bell N, Garland M (2009) Implementing sparse matrix-vector multiplication on throughput-oriented processors. In: SC ’09: proceedings of the conference on high performance computing networking, storage and analysis, pp 1–11.
Crouzeix M, Raviart P-A (1973) Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. Revue française d’automatique, informatique, recherche opérationnelle. Mathématique 7:33–76
van Dyk D, Geveler M, Mallach S, Ribbrock D, Gddeke D, Gutwenger C (2009) HONEI: a collection of libraries for numerical computations targeting multiple processor architectures. Comput Phys Commun 180(12):2534–2543
FeatFlow http://www.featflow.de. Accessed 4 Dec 2012
Fletcher CAJ (1983) The group finite element formulation. Comput Methods Appl Mech Eng 37:225–243
Gallouët T, Gastaldo L, Herbin R, Latché J-C (2008) An unconditionally stable pressure correction scheme for the compressible barotropic Navier–Stokes equations. ESAIM Math Model Numer Anal 42:303–331. doi:10.1051/mzan:2008005
Georgiev I, Kraus J, Margenov S (2008) Multilevel preconditioning of rotated bilinear non-conforming FEM problems. Comput Math Appl 55:2280–2294
Grimes R, Kincaid D, Young D (1979) ITPACK 2.0 users guide. Technical Report CNA-150, Center for Numerical Analysis, University of Texas. http://rene.ma.utexas.edu/CNA/ITPACK/
Jameson A (1993) Computational algorithms for aerodynamic analysis and design. Appl Numer Math 13:33–422
John V (1997) Parallele Lösung der inkompressiblen Navier–Stokes Gleichungen auf adaptiv verfeinerten Gittern. PhD thesis, Otto-von-Guericke Universität Magdeburg
Kang KS (2003) \(P_1\) nonconforming finite element methods for the solution of radiation transport problems. SIAM J Sci Comput 25(2):369–384
Kuzmin D (2001) Positive finite element schemes based on the flux-corrected transport procedure. In: Bathe KJ (ed) Computational fluid and solid mechanics. Elsevier, Amsterdam, pp 887–888
Kuzmin D (2006) On the design of general-purpose flux limiters for implicit FEM with a consistent mass matrix. I. Scalar convection. J Comput Phys 219:513–531
Kuzmin D (2008) On the design of algebraic flux correction schemes for quadratic finite elements. Comput Appl Math 218:79–87
Kuzmin D (2009) Explicit and implicit FEM-FCT algorithms with flux linearization. J Comput Phys 228:2517–2534
Kuzmin D (2010) A guide to numerical methods for transport equations. University Erlangen-Nuremberg, Erlangen. http://www.mathematik.uni-dortmund.de/~kuzmin/Transport.pdf
Kuzmin D (2012) Algebraic flux correction I. Scalar conservation laws. In: Kuzmin D, Löhner R, Turek S (eds) Flux-corrected transport: principles, algorithms, and applications, 2nd edn. Springer, Berlin, pp 145–192
Kuzmin D (2012) Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes. J Comput Appl Math 236:2317–2337
Kuzmin D, Möller M, Turek S (2004) High-resolution FEM-FCT schemes for multidimensional conservation laws. Comput Methods Appl Mech Eng 193:4915–4946
Kuzmin D, Möller M (2005) Algebraic flux correction I. Scalar conservation laws. In: Kuzmin D, Löhner R, Turek S (eds) Flux-corrected transport: principles, algorithms, and applications. Springer, Berlin
Kuzmin D, Turek S (2002) Flux correction tools for finite elements. J Comput Phys 175:525–558
Kuzmin D, Möller M, Turek S (2003) Multidimensional FEM-FCT schemes for arbitrary time-stepping. Int J Numer Methods Fluids 42:265–295
Kuzmin D, Turek S (2004) High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. J Comput Phys 198:131–158
Lapin A (2001) University of Stuttgart. Private communication
Layton WJ, Maubach JM, Rabier PJ (1998) Robustness of an elementwise parallel finite element method for convection–diffusion problems. SIAM J Sci Comput 19(6):1870–1891
LeVeque RJ (1996) High-resolution conservative algorithms for advection in incompressible flow. SIAM J Numer Anal 33:627–665
Möller M (2012) On the design of non-conforming high-resolution finite element schemes. In: Eberhardsteiner J et al (eds) Proceedings of the 6th European congress on computational methods in applied sciences and engineering (ECCOMAS 2012), Vienna
Rannacher R, Turek S (1992) A simple nonconforming quadrilateral Stokes element. Numer Methods PDEs 8:97–111
Schieweck F (1997) Parallele Lösung der stationären inkompressiblen Navier–Stokes Gleichungen. Habilitation thesis, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik
Turek S (1992) On ordering strategies in a multigrid algorithm. In: Proceedings of 8th GAMM-seminar. Notes on numerical fluid mechanics, vol 41
Vázquez F, Fernández JJ, Garzón EM (2012) Automatic tuning of the sparse matrix vector product on GPUs based on the ELLR-T approach. Parallel Comput 38:408–420
Williams S, Oliker L, Vuduc R, Shalf J, Yelick K, Demmel J (2007) Optimization of sparse matrix-vector multiplication on emerging multicore platforms. In: Proceedings of the (2007) ACM/IEEE conference on supercomputing. ACM, pp 38:1–38:12
Williams S, Bell N, Choi JW, Garland M, Oliker L, Vuduc R (2011) Sparse matrix-vector multiplication on multicore and accelerators. In: Kurzak J, Dongarra JJ, Bader DA (eds) Scientific computing with multicore and accelerators. CRC Press, Boca Raton, pp 83–109
Zalesak ST (1979) Fully multidimensional flux-corrected transport algorithms for fluids. J Comput Phys 31:335–362
Acknowledgments
The author wishes to thank professor Friedhelm Schieweck from Otto-von-Guericke Universität Magdeburg for fruitful discussions on nonconforming finite element approximations. He is also very thankful to the anonymous reviewers for giving helpful comments and for sharing their opinion on the restricted usefulness of nonconforming AFC-type methods. Finally, the author expresses gratitude to his colleagues Dipl-Infs. Markus Geveler and Dirk Ribbrock for providing benchmark results for SpMV implementations on GPUs. This work was supported by the German Research Foundation (DFG) under Grant SFB 708 “3D-Surface Engineering of Tools for Sheet Metal Forming—Manufacturing, Modelling, Machining”.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Möller, M. Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems. Computing 95, 425–448 (2013). https://doi.org/10.1007/s00607-012-0276-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00607-012-0276-y
Keywords
- Non-/conforming finite elements
- High-resolution schemes
- Flux-corrected transport
- Hyperbolic problems
- Convection-dominated flows