## Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems.(English)Zbl 1314.76034

Computing 95, No. 5, 425-448 (2013); erratum ibid. 96, No. 11, 1111-1112 (2014).
Summary: This paper is concerned with the extension of the algebraic flux-correction (AFC) approach 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 [M. Crouzeix and P. A. Raviart, Rev. Franc. Automat. Inform. Rech. Operat., R 7, No. 3, 33–76 (1974; Zbl 0302.65087)] on triangular meshes and its quadrilateral counterpart, the class of rotated bilinear Rannacher-Turek elements [R. Rannacher and S. Turek, Numer. Methods Partial Differ. Equations 8, No. 2, 97–111 (1992; Zbl 0742.76051)]. 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.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65Y20 Complexity and performance of numerical algorithms

### Citations:

Zbl 0302.65087; Zbl 0742.76051

### Software:

ITPACK 2C; ITPACK; FEATFLOW; HONEI
Full Text:

### References:

 [1] Anderson, DG, Iterative procedure for nonlinear integral equations, J Assoc Comput Mach, 12, 547-560, (1965) · Zbl 0149.11503 [2] 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. · Zbl 0916.65107 [3] 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 · Zbl 0852.76057 [4] Dyk, D; Geveler, M; Mallach, S; Ribbrock, D; Gddeke, D; Gutwenger, C, HONEI: a collection of libraries for numerical computations targeting multiple processor architectures, Comput Phys Commun, 180, 2534-2543, (2009) · Zbl 1197.65007 [5] FeatFlow http://www.featflow.de. Accessed 4 Dec 2012 · Zbl 1055.76029 [6] Fletcher, CAJ, The group finite element formulation, Comput Methods Appl Mech Eng, 37, 225-243, (1983) [7] 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 · Zbl 1132.35433 [8] Georgiev, I; Kraus, J; Margenov, S, Multilevel preconditioning of rotated bilinear non-conforming FEM problems, Comput Math Appl, 55, 2280-2294, (2008) · Zbl 1142.65323 [9] 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/ · Zbl 1241.65083 [10] Jameson, A, Computational algorithms for aerodynamic analysis and design, Appl Numer Math, 13, 33-422, (1993) · Zbl 0792.76049 [11] John V (1997) Parallele Lösung der inkompressiblen Navier-Stokes Gleichungen auf adaptiv verfeinerten Gittern. PhD thesis, Otto-von-Guericke Universität Magdeburg [12] Kang, KS, $$P_1$$ nonconforming finite element methods for the solution of radiation transport problems, SIAM J Sci Comput, 25, 369-384, (2003) · Zbl 1042.65073 [13] Kuzmin, D; Bathe, KJ (ed.), Positive finite element schemes based on the flux-corrected transport procedure, 887-888, (2001), Amsterdam [14] Kuzmin, D, 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, (2006) · Zbl 1189.76342 [15] Kuzmin, D, On the design of algebraic flux correction schemes for quadratic finite elements, Comput Appl Math, 218, 79-87, (2008) · Zbl 1143.65092 [16] Kuzmin, D, Explicit and implicit FEM-FCT algorithms with flux linearization, J Comput Phys, 228, 2517-2534, (2009) · Zbl 1275.76171 [17] Kuzmin D (2010) A guide to numerical methods for transport equations. University Erlangen-Nuremberg, Erlangen. http://www.mathematik.uni-dortmund.de/ kuzmin/Transport.pdf [18] Kuzmin, D; Kuzmin, D (ed.); Löhner, R (ed.); Turek, S (ed.), Algebraic flux correction I. scalar conservation laws, 145-192, (2012), Berlin [19] Kuzmin, D, Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes, J Comput Appl Math, 236, 2317-2337, (2012) · Zbl 1241.65083 [20] Kuzmin, D; Möller, M; Turek, S, High-resolution FEM-FCT schemes for multidimensional conservation laws, Comput Methods Appl Mech Eng, 193, 4915-4946, (2004) · Zbl 1112.76393 [21] Kuzmin, D; Möller, M; Kuzmin, D (ed.); Löhner, R (ed.); Turek, S (ed.), Algebraic flux correction I. scalar conservation laws, (2005), Berlin · Zbl 1062.76003 [22] Kuzmin, D; Turek, S, Flux correction tools for finite elements, J Comput Phys, 175, 525-558, (2002) · Zbl 1028.76023 [23] Kuzmin, D; Möller, M; Turek, S, Multidimensional FEM-FCT schemes for arbitrary time-stepping, Int J Numer Methods Fluids, 42, 265-295, (2003) · Zbl 1055.76029 [24] Kuzmin, D; Turek, S, High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter, J Comput Phys, 198, 131-158, (2004) · Zbl 1107.76352 [25] Lapin A (2001) University of Stuttgart. Private communication [26] Layton, WJ; Maubach, JM; Rabier, PJ, Robustness of an elementwise parallel finite element method for convection-diffusion problems, SIAM J Sci Comput, 19, 1870-1891, (1998) · Zbl 0916.65107 [27] LeVeque, RJ, High-resolution conservative algorithms for advection in incompressible flow, SIAM J Numer Anal, 33, 627-665, (1996) · Zbl 0852.76057 [28] 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 · Zbl 0416.76002 [29] Rannacher, R; Turek, S, A simple nonconforming quadrilateral Stokes element, Numer Methods PDEs, 8, 97-111, (1992) · Zbl 0742.76051 [30] Schieweck F (1997) Parallele Lösung der stationären inkompressiblen Navier-Stokes Gleichungen. Habilitation thesis, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik · Zbl 0915.76051 [31] Turek S (1992) On ordering strategies in a multigrid algorithm. In: Proceedings of 8th GAMM-seminar. Notes on numerical fluid mechanics, vol 41 · Zbl 0787.76047 [32] Vázquez, F; Fernández, JJ; Garzón, EM, Automatic tuning of the sparse matrix vector product on GPUs based on the ELLR-T approach, Parallel Comput, 38, 408-420, (2012) [33] 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 · Zbl 1055.76029 [34] Williams, S; Bell, N; Choi, JW; Garland, M; Oliker, L; Vuduc, R; Kurzak, J (ed.); Dongarra, JJ (ed.); Bader, DA (ed.), Sparse matrix-vector multiplication on multicore and accelerators, 83-109, (2011), Boca Raton [35] Zalesak, ST, Fully multidimensional flux-corrected transport algorithms for fluids, J Comput Phys, 31, 335-362, (1979) · Zbl 0416.76002
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