On the solvability of the nonlinear problems in an algebraically stabilized finite element method for evolutionary transport-dominated equations

John, Volker; Knobloch, Petr; Korsmeier, Paul

doi:10.1090/mcom/3576

On the solvability of the nonlinear problems in an algebraically stabilized finite element method for evolutionary transport-dominated equations
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Math. Comp. 90 (2021), 595-611 Request permission

Abstract:

The so-called FEM-FCT (finite element method flux-corrected transport) scheme for evolutionary scalar convection-dominated equations leads in each time instant to a nonlinear problem. For sufficiently small time steps, the existence and uniqueness of a solution of these problems is shown. Moreover, the convergence of a semi-smooth Newton’s method is studied.

References

Santiago Badia and Jesús Bonilla, Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization, Comput. Methods Appl. Mech. Engrg. 313 (2017), 133–158. MR 3577934, DOI 10.1016/j.cma.2016.09.035
Santiago Badia and Alba Hierro, On monotonicity-preserving stabilized finite element approximations of transport problems, SIAM J. Sci. Comput. 36 (2014), no. 6, A2673–A2697. MR 3278847, DOI 10.1137/130927206
Gabriel R. Barrenechea, Erik Burman, and Fotini Karakatsani, Edge-based nonlinear diffusion for finite element approximations of convection-diffusion equations and its relation to algebraic flux-correction schemes, Numer. Math. 135 (2017), no. 2, 521–545. MR 3599568, DOI 10.1007/s00211-016-0808-z
Gabriel R. Barrenechea, Volker John, and Petr Knobloch, Some analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in one dimension, IMA J. Numer. Anal. 35 (2015), no. 4, 1729–1756. MR 3407242, DOI 10.1093/imanum/dru041
Gabriel R. Barrenechea, Volker John, and Petr Knobloch, Analysis of algebraic flux correction schemes, SIAM J. Numer. Anal. 54 (2016), no. 4, 2427–2451. MR 3537011, DOI 10.1137/15M1018216
Gabriel R. Barrenechea, Volker John, and Petr Knobloch, An algebraic flux correction scheme satisfying the discrete maximum principle and linearity preservation on general meshes, Math. Models Methods Appl. Sci. 27 (2017), no. 3, 525–548. MR 3621336, DOI 10.1142/S0218202517500087
Gabriel R. Barrenechea, Volker John, Petr Knobloch, and Richard Rankin, A unified analysis of algebraic flux correction schemes for convection-diffusion equations, SeMA J. 75 (2018), no. 4, 655–685. MR 3875919, DOI 10.1007/s40324-018-0160-6
J. P. Boris and D. L. Book, Flux-corrected transport. I: SHASTA, a fluid transport algorithm that works, J. Comput. Phys. 11 (1973), 38–69.
Erik Burman, A monotonicity preserving, nonlinear, finite element upwind method for the transport equation, Appl. Math. Lett. 49 (2015), 141–146. MR 3361708, DOI 10.1016/j.aml.2015.05.005
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
Jean-Luc Guermond and Murtazo Nazarov, A maximum-principle preserving $C^0$ finite element method for scalar conservation equations, Comput. Methods Appl. Mech. Engrg. 272 (2014), 198–213. MR 3171280, DOI 10.1016/j.cma.2013.12.015
Volker John, Petr Knobloch, and Julia Novo, Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?, Comput. Vis. Sci. 19 (2018), no. 5-6, 47–63. MR 3893442, DOI 10.1007/s00791-018-0290-5
Volker John and Julia Novo, On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations, J. Comput. Phys. 231 (2012), no. 4, 1570–1586. MR 2876468, DOI 10.1016/j.jcp.2011.10.025
Volker John and Ellen Schmeyer, Finite element methods for time-dependent convection-diffusion-reaction equations with small diffusion, Comput. Methods Appl. Mech. Engrg. 198 (2008), no. 3-4, 475–494. MR 2479278, DOI 10.1016/j.cma.2008.08.016
D. Kuzmin, Positive element schemes based on the flux-corrected transport procedure, Computational Fluid and Solid Mechanics (K.J. Bathe, ed.), Elsevier, 2001, pp. 887–888.
D. Kuzmin and S. Turek, Flux correction tools for finite elements, J. Comput. Phys. 175 (2002), no. 2, 525–558. MR 1880117, DOI 10.1006/jcph.2001.6955
Dmitri Kuzmin, Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes, J. Comput. Appl. Math. 236 (2012), no. 9, 2317–2337. MR 2879702, DOI 10.1016/j.cam.2011.11.019
Dmitri Kuzmin, Steffen Basting, and John N. Shadid, Linearity-preserving monotone local projection stabilization schemes for continuous finite elements, Comput. Methods Appl. Mech. Engrg. 322 (2017), 23–41. MR 3660774, DOI 10.1016/j.cma.2017.04.030
Dmitri Kuzmin and Matthias Möller, Algebraic flux correction. I. Scalar conservation laws, Flux-corrected transport, Sci. Comput., Springer, Berlin, 2005, pp. 155–206. MR 2129255, DOI 10.1007/3-540-27206-2_{6}
R. Löhner, K. Morgan, J. Peraire, and M. Vahdati, Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier-Stokes equations, Int. J. Numer. Methods Fluids 7 (1987), 1093–1109.
M. Möller, D. Kuzmin, and D. Kourounis, Implicit FEM-FCT algorithms and discrete Newton methods for transient convection problems, Internat. J. Numer. Methods Fluids 57 (2008), no. 6, 761–792. MR 2422132, DOI 10.1002/fld.1654
Hans-Görg Roos, Martin Stynes, and Lutz Tobiska, Robust numerical methods for singularly perturbed differential equations, 2nd ed., Springer Series in Computational Mathematics, vol. 24, Springer-Verlag, Berlin, 2008. Convection-diffusion-reaction and flow problems. MR 2454024
Stefan Scholtes, Introduction to piecewise differentiable equations, SpringerBriefs in Optimization, Springer, New York, 2012. MR 2953259, DOI 10.1007/978-1-4614-4340-7
Michael Ulbrich, Semismooth Newton methods for variational inequalities and constrained optimization problems in function spaces, MOS-SIAM Series on Optimization, vol. 11, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA, 2011. MR 2839219, DOI 10.1137/1.9781611970692
Steven T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys. 31 (1979), no. 3, 335–362. MR 534786, DOI 10.1016/0021-9991(79)90051-2