A unified analysis of algebraic flux correction schemes for convection–diffusion equations

Ainsworth, M., Allendes, A., Barrenechea, G.R., Rankin, R.: Fully computable a posteriori error bounds for stabilised FEM approximations of convection–reaction–diffusion problems in three dimensions. Int. J. Numer. Methods Fluids 73(9), 765–790 (2013)

Allendes, A., Barrenechea, G.R., Rankin, R.: Fully computable error estimation of a nonlinear, positivity-preserving discretization of the convection–diffusion–reaction equation. SIAM J. Sci. Comput. 39(5), A1903–A1927 (2017)

Anderson, D.G.: Iterative procedures for nonlinear integral equations. J. Assoc. Comput. Mach. 12, 547–560 (1965)

Arminjon, P., Dervieux, A.: Construction of TVD-like artificial viscosities on two-dimensional arbitrary FEM grids. J. Comput. Phys. 106(1), 176–198 (1993)

Augustin, M., Caiazzo, A., Fiebach, A., Fuhrmann, J., John, V., Linke, A., Umla, R.: An assessment of discretizations for convection-dominated convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 200(47–48), 3395–3409 (2011)

Badia, S., Bonilla, J.: Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization. Comput. Methods Appl. Mech. Eng. 313, 133–158 (2017)

Badia, S., Hierro, A.: On monotonicity-preserving stabilized finite element approximations of transport problems. SIAM J. Sci. Comput. 36(6), A2673–A2697 (2014)

Barrenechea, G.R., John, V., Knobloch, P.: Some analytical results for an algebraic flux correction scheme for a steady convection–diffusion equation in one dimension. IMA J. Numer. Anal. 35(4), 1729–1756 (2015)

Barrenechea, G.R., John, V., Knobloch, P.: Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal. 54(4), 2427–2451 (2016)

Barrenechea, G.R., Burman, E., Karakatsani, F.: Edge-based nonlinear diffusion for finite element approximations of convection–diffusion equations and its relation to algebraic flux-correction schemes. Numer. Math. 135(2), 521–545 (2017)

Barrenechea, G.R., John, V., Knobloch, P.: An algebraic flux correction scheme satisfying the discrete maximum principle and linearity preservation on general meshes. Math. Models Methods Appl. Sci. 27(3), 525–548 (2017)

Becker, R., Braack, M.: A two-level stabilization scheme for the Navier–Stokes equations. In: Feistauer, M., Dolejší, V., Knobloch, P., Najzar, K. (eds.) Numerical Mathematics and Advanced Applications, pp. 123–130. Springer, Berlin (2004)

Bordás, R., John, V., Schmeyer, E., Thévenin, D.: Numerical methods for the simulation of a coalescence-driven droplet size distribution. Theor. Comput. Fluid Dyn. 27(3–4), 253–271 (2013)

Boris, J.P., Book, D.L.: Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 11(1), 38–69 (1973)

Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov–Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3), 199–259 (1982). FENOMECH ’81, Part I (Stuttgart, 1981)

Burman, E., Hansbo, P.: Edge stabilization for Galerkin approximations of convection–diffusion–reaction problems. Comput. Methods Appl. Mech. Eng. 193(15–16), 1437–1453 (2004)

Davis, T.A.: Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30(2), 196–199 (2004)

Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159. Springer, New York (2004)

Guermond, J.-L., Popov, B.: Invariant domains and second-order continuous finite element approximation for scalar conservation equations. SIAM J. Numer. Anal. 55(6), 3120–3146 (2017)

Guermond, J.-L., Nazarov, M., Popov, B., Yang, Y.: A second-order maximum principle preserving Lagrange finite element technique for nonlinear scalar conservation equations. SIAM J. Numer. Anal. 52(4), 2163–2182 (2014)

Hemker, P.W.: A singularly perturbed model problem for numerical computation. J. Comput. Appl. Math. 76(1–2), 277–285 (1996)

Hughes, T.J.R., Brooks, A.: A multidimensional upwind scheme with no crosswind diffusion. In: Finite Element Methods for Convection Dominated Flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 1979), AMD, vol. 34, pp. 19–35. Amer. Soc. Mech. Engrs. (ASME), New York (1979)

Jameson, A.: Origins and further development of the Jameson–Schmidt–Turkel scheme. AIAA J. 55, 1487–1510 (2017)

Jameson, A., Schmidt, W., Turkel, E.: Numerical solution of the Euler equations by finite volume methods using Runge–Kutta time-stepping schemes. AIAA J. In: 14th AIAA Fluid and Plasma Dynamics Conference, AIAA paper 1981-1259 (1981)

John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations. I. A review. Comput. Methods Appl. Mech. Eng. 196(17–20), 2197–2215 (2007)

John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations. II. Analysis for $$P_1$$ P 1 and $$Q_1$$ Q 1 finite elements. Comput. Methods Appl. Mech. Eng. 197(21–24), 1997–2014 (2008)

John, V., Matthies, G.: MooNMD—a program package based on mapped finite element methods. Comput. Vis. Sci. 6(2–3), 163–169 (2004)

John, V., Novo, J.: On (essentially) non-oscillatory discretizations of evolutionary convection–diffusion equations. J. Comput. Phys. 231(4), 1570–1586 (2012)

John, V., Schmeyer, E.: Finite element methods for time-dependent convection–diffusion–reaction equations with small diffusion. Comput. Methods Appl. Mech. Eng. 198(3–4), 475–494 (2008)

John, V., Knobloch, P., Novo, J.: Finite elements for scalar convection-dominated equations and incompressible flow problems—a never ending story? Comput. Vis. Sci. (2018). https://doi.org/10.1007/s00791-018-0290-5

Knobloch, P.: Numerical solution of convection–diffusion equations using a nonlinear method of upwind type. J. Sci. Comput. 43(3), 454–470 (2010)

Knobloch, P.: On the discrete maximum principle for algebraic flux correction schemes with limiters of upwind type. In: Zhang, Z., Huang, Z., Stynes, M. (eds.) Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2016, Lecture Notes in Computational Science and Engineering, vol. 120, pp. 129–139. Springer, New York (2017)

Kuzmin, D.: On the design of general-purpose flux limiters for finite element schemes. I. Scalar convection. J. Comput. Phys. 219(2), 513–531 (2006)

Kuzmin, D.: Algebraic flux correction for finite element discretizations of coupled systems. In: Papadrakakis, M., Oñate, E., Schrefler, B. (eds.) Proceedings of the International Conference on Computational Methods for Coupled Problems in Science and Engineering, pp. 1–5. CIMNE, Barcelona (2007)

Kuzmin, D.: Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes. J. Comput. Appl. Math. 236(9), 2317–2337 (2012)

Kuzmin, D., Möller, M.: Algebraic flux correction I. Scalar conservation laws. In: Kuzmin, D., Löhner, R., Turek, S. (eds.) Flux-Corrected Transport. Principles, Algorithms, and Applications, pp. 155–206. Springer, Berlin (2005)

Kuzmin, D., Turek, S.: High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. J. Comput. Phys. 198(1), 131–158 (2004)

Lohmann, C., Kuzmin, D., Shadid, J.N., Mabuza, S.: Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements. J. Comput. Phys. 344, 151–186 (2017)

Löhner, R., Morgan, K., Peraire, J., Vahdati, M.: Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier–Stokes equations. Int. J. Numer. Methods Fluids 7(10), 1093–1109 (1987)

Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and its Applications, vol. 2. North-Holland, Amsterdam, (1977)

Walker, H.F., Ni, P.: Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal. 49(4), 1715–1735 (2011)

Wesseling, P.: Principles of Computational Fluid Dynamics, Springer Series in Computational Mathematics, vol. 29. Springer, Berlin (2001)

Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31(3), 335–362 (1979)