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Erschienen in: Computing and Visualization in Science 5-6/2018

05.02.2018 | Special Issue FEM Symposium 2017

Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?

verfasst von: Volker John, Petr Knobloch, Julia Novo

Erschienen in: Computing and Visualization in Science | Ausgabe 5-6/2018

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Abstract

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important open problems in these fields are discussed. The exposition concentrates on \(H^1\)-conforming finite elements.

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Literatur
1.
Zurück zum Zitat Acosta, G., Durán, R.G.: The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J. Numer. Anal. 37(1), 18–36 (1999)MathSciNetMATH Acosta, G., Durán, R.G.: The maximum angle condition for mixed and nonconforming elements: application to the Stokes equations. SIAM J. Numer. Anal. 37(1), 18–36 (1999)MathSciNetMATH
2.
Zurück zum Zitat Ahmed, N., Bartsch, C., John, V., Wilbrandt, U.: An Assessment of Some Solvers for Saddle Point Problems Emerging from the Incompressible Navier–Stokes Equations. Comput. Methods Appl. Mech. Eng. 331, 492–513 (2018)MathSciNet Ahmed, N., Bartsch, C., John, V., Wilbrandt, U.: An Assessment of Some Solvers for Saddle Point Problems Emerging from the Incompressible Navier–Stokes Equations. Comput. Methods Appl. Mech. Eng. 331, 492–513 (2018)MathSciNet
3.
Zurück zum Zitat Ainsworth, M., Barrenechea, G.R., Wachtel, A.: Stabilization of high aspect ratio mixed finite elements for incompressible flow. SIAM J. Numer. Anal. 53(2), 1107–1120 (2015)MathSciNetMATH Ainsworth, M., Barrenechea, G.R., Wachtel, A.: Stabilization of high aspect ratio mixed finite elements for incompressible flow. SIAM J. Numer. Anal. 53(2), 1107–1120 (2015)MathSciNetMATH
4.
Zurück zum Zitat Ainsworth, M., Coggins, P.: The stability of mixed \(hp\)-finite element methods for Stokes flow on high aspect ratio elements. SIAM J. Numer. Anal. 38(5), 1721–1761 (2000)MathSciNetMATH Ainsworth, M., Coggins, P.: The stability of mixed \(hp\)-finite element methods for Stokes flow on high aspect ratio elements. SIAM J. Numer. Anal. 38(5), 1721–1761 (2000)MathSciNetMATH
5.
Zurück zum Zitat Allendes, A., Durán, F., Rankin, R.: Error estimation for low-order adaptive finite element approximations for fluid flow problems. IMA J. Numer. Anal. 36(4), 1715–1747 (2016)MathSciNetMATH Allendes, A., Durán, F., Rankin, R.: Error estimation for low-order adaptive finite element approximations for fluid flow problems. IMA J. Numer. Anal. 36(4), 1715–1747 (2016)MathSciNetMATH
6.
Zurück zum Zitat Apel, T., Knopp, T., Lube, G.: Stabilized finite element methods with anisotropic mesh refinement for the Oseen problem. Appl. Numer. Math. 58(12), 1830–1843 (2008)MathSciNetMATH Apel, T., Knopp, T., Lube, G.: Stabilized finite element methods with anisotropic mesh refinement for the Oseen problem. Appl. Numer. Math. 58(12), 1830–1843 (2008)MathSciNetMATH
7.
Zurück zum Zitat Apel, T., Randrianarivony, H.M.: Stability of discretizations of the Stokes problem on anisotropic meshes. Math. Comput. Simul. 61(3–6), 437–447 (2003)MathSciNetMATH Apel, T., Randrianarivony, H.M.: Stability of discretizations of the Stokes problem on anisotropic meshes. Math. Comput. Simul. 61(3–6), 437–447 (2003)MathSciNetMATH
8.
Zurück zum Zitat Apel, T., Matthies, G.: Nonconforming, anisotropic, rectangular finite elements of arbitrary order for the Stokes problem. SIAM J. Numer. Anal. 46(4), 1867–1891 (2008)MathSciNetMATH Apel, T., Matthies, G.: Nonconforming, anisotropic, rectangular finite elements of arbitrary order for the Stokes problem. SIAM J. Numer. Anal. 46(4), 1867–1891 (2008)MathSciNetMATH
9.
Zurück zum Zitat Apel, T., Nicaise, S.: The inf-sup condition for low order elements on anisotropic meshes. Calcolo 41(2), 89–113 (2004)MathSciNetMATH Apel, T., Nicaise, S.: The inf-sup condition for low order elements on anisotropic meshes. Calcolo 41(2), 89–113 (2004)MathSciNetMATH
10.
Zurück zum Zitat Apel, T., Nicaise, S., Schöberl, J.: A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges. IMA J. Numer. Anal. 21(4), 843–856 (2001)MathSciNetMATH Apel, T., Nicaise, S., Schöberl, J.: A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges. IMA J. Numer. Anal. 21(4), 843–856 (2001)MathSciNetMATH
11.
Zurück zum Zitat Arminjon, P., Dervieux, A.: Construction of TVD-like artificial viscosities on two-dimensional arbitrary FEM grids. J. Comput. Phys. 106(1), 176–198 (1993)MathSciNetMATH Arminjon, P., Dervieux, A.: Construction of TVD-like artificial viscosities on two-dimensional arbitrary FEM grids. J. Comput. Phys. 106(1), 176–198 (1993)MathSciNetMATH
12.
Zurück zum Zitat Arndt, D., Dallmann, H., Lube, G.: Local projection FEM stabilization for the time-dependent incompressible Navier–Stokes problem. Numer. Methods Part. Differ. Equ. 31(4), 1224–1250 (2015)MathSciNetMATH Arndt, D., Dallmann, H., Lube, G.: Local projection FEM stabilization for the time-dependent incompressible Navier–Stokes problem. Numer. Methods Part. Differ. Equ. 31(4), 1224–1250 (2015)MathSciNetMATH
13.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
14.
15.
Zurück zum Zitat Bardos, C.W., Titi, E.S.: Mathematics and turbulence: where do we stand? J. Turbul. 14(3), 42–76 (2013)MathSciNet Bardos, C.W., Titi, E.S.: Mathematics and turbulence: where do we stand? J. Turbul. 14(3), 42–76 (2013)MathSciNet
16.
Zurück zum Zitat Barrenechea, G.R., John, V., Knobloch, P.: A local projection stabilization finite element method with nonlinear crosswind diffusion for convection–diffusion–reaction equations. ESAIM Math. Model. Numer. Anal. 47(5), 1335–1366 (2013)MathSciNetMATH Barrenechea, G.R., John, V., Knobloch, P.: A local projection stabilization finite element method with nonlinear crosswind diffusion for convection–diffusion–reaction equations. ESAIM Math. Model. Numer. Anal. 47(5), 1335–1366 (2013)MathSciNetMATH
17.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
18.
Zurück zum Zitat Barrenechea, G.R., John, V., Knobloch, P.: Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal. 54(4), 2427–2451 (2016)MathSciNetMATH Barrenechea, G.R., John, V., Knobloch, P.: Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal. 54(4), 2427–2451 (2016)MathSciNetMATH
19.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
20.
Zurück zum Zitat Barrenechea, G.R., Valentin, F.: Consistent local projection stabilized finite element methods. SIAM J. Numer. Anal. 48(5), 1801–1825 (2010)MathSciNetMATH Barrenechea, G.R., Valentin, F.: Consistent local projection stabilized finite element methods. SIAM J. Numer. Anal. 48(5), 1801–1825 (2010)MathSciNetMATH
21.
Zurück zum Zitat Barrenechea, G.R., Valentin, F.: A residual local projection method for the Oseen equation. Comput. Methods Appl. Mech. Eng. 199(29–32), 1906–1921 (2010)MathSciNetMATH Barrenechea, G.R., Valentin, F.: A residual local projection method for the Oseen equation. Comput. Methods Appl. Mech. Eng. 199(29–32), 1906–1921 (2010)MathSciNetMATH
22.
Zurück zum Zitat Barrenechea, G.R., Valentin, F.: Beyond pressure stabilization: a low-order local projection method for the Oseen equation. Int. J. Numer. Methods Eng. 86(7), 801–815 (2011)MathSciNetMATH Barrenechea, G.R., Valentin, F.: Beyond pressure stabilization: a low-order local projection method for the Oseen equation. Int. J. Numer. Methods Eng. 86(7), 801–815 (2011)MathSciNetMATH
23.
Zurück zum Zitat Barrios, T.P., Cascón, J.M., González, M.: Augmented mixed finite element method for the Oseen problem: a priori and a posteriori error analyses. Comput. Methods Appl. Mech. Eng. 313, 216–238 (2017)MathSciNet Barrios, T.P., Cascón, J.M., González, M.: Augmented mixed finite element method for the Oseen problem: a priori and a posteriori error analyses. Comput. Methods Appl. Mech. Eng. 313, 216–238 (2017)MathSciNet
24.
Zurück zum Zitat Bazilevs, Y., Beirão da Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes. Math. Models Methods Appl. Sci. 16(7), 1031–1090 (2006)MathSciNetMATH Bazilevs, Y., Beirão da Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes. Math. Models Methods Appl. Sci. 16(7), 1031–1090 (2006)MathSciNetMATH
25.
Zurück zum Zitat Bazilevs, Y., Calo, V.M., Tezduyar, T.E., Hughes, T.J.R.: \(YZ\beta \) discontinuity capturing for advection-dominated processes with application to arterial drug delivery. Int. J. Numer. Methods Fluids 54(6–8), 593–608 (2007)MathSciNetMATH Bazilevs, Y., Calo, V.M., Tezduyar, T.E., Hughes, T.J.R.: \(YZ\beta \) discontinuity capturing for advection-dominated processes with application to arterial drug delivery. Int. J. Numer. Methods Fluids 54(6–8), 593–608 (2007)MathSciNetMATH
26.
Zurück zum Zitat Becker, R., Braack, M.: A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38(4), 173–199 (2001)MathSciNetMATH Becker, R., Braack, M.: A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38(4), 173–199 (2001)MathSciNetMATH
27.
Zurück zum Zitat 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) 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)
28.
Zurück zum Zitat Benzi, M., Olshanskii, M.A.: An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput. 28(6), 2095–2113 (2006)MathSciNetMATH Benzi, M., Olshanskii, M.A.: An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput. 28(6), 2095–2113 (2006)MathSciNetMATH
29.
Zurück zum Zitat Benzi, M., Wang, Z.: Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier–Stokes equations. SIAM J. Sci. Comput. 33(5), 2761–2784 (2011)MathSciNetMATH Benzi, M., Wang, Z.: Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier–Stokes equations. SIAM J. Sci. Comput. 33(5), 2761–2784 (2011)MathSciNetMATH
30.
Zurück zum Zitat Berrone, S.: Robustness in a posteriori error analysis for FEM flow models. Numer. Math. 91(3), 389–422 (2002)MathSciNetMATH Berrone, S.: Robustness in a posteriori error analysis for FEM flow models. Numer. Math. 91(3), 389–422 (2002)MathSciNetMATH
31.
Zurück zum Zitat Bochev, P., Gunzburger, M.: An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations. SIAM J. Numer. Anal. 42(3), 1189–1207 (2004)MathSciNetMATH Bochev, P., Gunzburger, M.: An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations. SIAM J. Numer. Anal. 42(3), 1189–1207 (2004)MathSciNetMATH
32.
Zurück zum Zitat 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)MATH 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)MATH
33.
Zurück zum Zitat Braack, M.: A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes. M2AN. Math. Model. Numer. Anal. 42(6), 903–924 (2008)MathSciNetMATH Braack, M.: A stabilized finite element scheme for the Navier-Stokes equations on quadrilateral anisotropic meshes. M2AN. Math. Model. Numer. Anal. 42(6), 903–924 (2008)MathSciNetMATH
34.
Zurück zum Zitat Braack, M., Burman, E., Taschenberger, N.: Duality based a posteriori error estimation for quasi-periodic solutions using time averages. SIAM J. Sci. Comput. 33(5), 2199–2216 (2011)MathSciNetMATH Braack, M., Burman, E., Taschenberger, N.: Duality based a posteriori error estimation for quasi-periodic solutions using time averages. SIAM J. Sci. Comput. 33(5), 2199–2216 (2011)MathSciNetMATH
35.
Zurück zum Zitat Braack, M., Lube, G., Röhe, L.: Divergence preserving interpolation on anisotropic quadrilateral meshes. Comput. Methods Appl. Math. 12(2), 123–138 (2012)MathSciNetMATH Braack, M., Lube, G., Röhe, L.: Divergence preserving interpolation on anisotropic quadrilateral meshes. Comput. Methods Appl. Math. 12(2), 123–138 (2012)MathSciNetMATH
36.
Zurück zum Zitat Braack, M., Mucha, P.B.: Directional do-nothing condition for the Navier-Stokes equations. J. Comput. Math. 32(5), 507–521 (2014)MathSciNetMATH Braack, M., Mucha, P.B.: Directional do-nothing condition for the Navier-Stokes equations. J. Comput. Math. 32(5), 507–521 (2014)MathSciNetMATH
37.
Zurück zum Zitat Brennecke, C., Linke, A., Merdon, C., Schöberl, J.: Optimal and pressure-independent \(L^2\) velocity error estimates for a modified Crouzeix–Raviart Stokes element with BDM reconstructions. J. Comput. Math. 33(2), 191–208 (2015)MathSciNetMATH Brennecke, C., Linke, A., Merdon, C., Schöberl, J.: Optimal and pressure-independent \(L^2\) velocity error estimates for a modified Crouzeix–Raviart Stokes element with BDM reconstructions. J. Comput. Math. 33(2), 191–208 (2015)MathSciNetMATH
38.
Zurück zum Zitat Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8(R–2), 129–151 (1974)MathSciNetMATH Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8(R–2), 129–151 (1974)MathSciNetMATH
39.
Zurück zum Zitat Brezzi, F., Fortin, M.: A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89(3), 457–491 (2001)MathSciNetMATH Brezzi, F., Fortin, M.: A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89(3), 457–491 (2001)MathSciNetMATH
40.
Zurück zum Zitat Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the Stokes equations. In: Efficient Solutions of Elliptic Systems (Kiel, 1984), Volume 10 of Notes Numer. Fluid Mech., pp. 11–19. Friedr. Vieweg, Braunschweig (1984) Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the Stokes equations. In: Efficient Solutions of Elliptic Systems (Kiel, 1984), Volume 10 of Notes Numer. Fluid Mech., pp. 11–19. Friedr. Vieweg, Braunschweig (1984)
41.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
42.
Zurück zum Zitat Buffa, A., de Falco, C., Sangalli, G.: IsoGeometric analysis: stable elements for the 2D Stokes equation. Int. J. Numer. Methods Fluids 65(11–12), 1407–1422 (2011)MathSciNetMATH Buffa, A., de Falco, C., Sangalli, G.: IsoGeometric analysis: stable elements for the 2D Stokes equation. Int. J. Numer. Methods Fluids 65(11–12), 1407–1422 (2011)MathSciNetMATH
43.
Zurück zum Zitat Bulling, J., John, V., Knobloch, P.: Isogeometric analysis for flows around a cylinder. Appl. Math. Lett. 63, 65–70 (2017)MathSciNetMATH Bulling, J., John, V., Knobloch, P.: Isogeometric analysis for flows around a cylinder. Appl. Math. Lett. 63, 65–70 (2017)MathSciNetMATH
44.
Zurück zum Zitat Burman, E.: A posteriori error estimation for interior penalty finite element approximations of the advection–reaction equation. SIAM J. Numer. Anal. 47(5), 3584–3607 (2009)MathSciNetMATH Burman, E.: A posteriori error estimation for interior penalty finite element approximations of the advection–reaction equation. SIAM J. Numer. Anal. 47(5), 3584–3607 (2009)MathSciNetMATH
45.
Zurück zum Zitat Burman, E.: Robust error estimates for stabilized finite element approximations of the two dimensional Navier–Stokes’ equations at high Reynolds number. Comput. Methods Appl. Mech. Eng. 288, 2–23 (2015)MathSciNetMATH Burman, E.: Robust error estimates for stabilized finite element approximations of the two dimensional Navier–Stokes’ equations at high Reynolds number. Comput. Methods Appl. Mech. Eng. 288, 2–23 (2015)MathSciNetMATH
46.
Zurück zum Zitat Burman, E., Ern, A.: Stabilized Galerkin approximation of convection–diffusion–reaction equations: discrete maximum principle and convergence. Math. Comput. 74(252), 1637–1652 (2005). (electronic)MathSciNetMATH Burman, E., Ern, A.: Stabilized Galerkin approximation of convection–diffusion–reaction equations: discrete maximum principle and convergence. Math. Comput. 74(252), 1637–1652 (2005). (electronic)MathSciNetMATH
47.
Zurück zum Zitat Burman, E., Ern, A., Fernández, M.A.: Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem. ESAIM: M2AN 51(2), 487–507 (2017)MathSciNetMATH Burman, E., Ern, A., Fernández, M.A.: Fractional-step methods and finite elements with symmetric stabilization for the transient Oseen problem. ESAIM: M2AN 51(2), 487–507 (2017)MathSciNetMATH
48.
Zurück zum Zitat Burman, E., Fernández, M.A.: Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence. Numer. Math. 107(1), 39–77 (2007)MathSciNetMATH Burman, E., Fernández, M.A.: Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence. Numer. Math. 107(1), 39–77 (2007)MathSciNetMATH
49.
Zurück zum Zitat Burman, E., Guzmán, J., Leykekhman, D.: Weighted error estimates of the continuous interior penalty method for singularly perturbed problems. IMA J. Numer. Anal. 29(2), 284–314 (2009)MathSciNetMATH Burman, E., Guzmán, J., Leykekhman, D.: Weighted error estimates of the continuous interior penalty method for singularly perturbed problems. IMA J. Numer. Anal. 29(2), 284–314 (2009)MathSciNetMATH
50.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
51.
Zurück zum Zitat Burman, E., Hansbo, P.: Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Eng. 195(19–22), 2393–2410 (2006)MathSciNetMATH Burman, E., Hansbo, P.: Edge stabilization for the generalized Stokes problem: a continuous interior penalty method. Comput. Methods Appl. Mech. Eng. 195(19–22), 2393–2410 (2006)MathSciNetMATH
52.
Zurück zum Zitat Burman, E., Santos, I.P.: Error estimates for transport problems with high Péclet number using a continuous dependence assumption. J. Comput. Appl. Math. 309, 267–286 (2017)MathSciNetMATH Burman, E., Santos, I.P.: Error estimates for transport problems with high Péclet number using a continuous dependence assumption. J. Comput. Appl. Math. 309, 267–286 (2017)MathSciNetMATH
53.
Zurück zum Zitat Charnyi, S., Heister, T., Olshanskii, M.A., Rebholz, L.G.: On conservation laws of Navier–Stokes Galerkin discretizations. J. Comput. Phys. 337, 289–308 (2017)MathSciNet Charnyi, S., Heister, T., Olshanskii, M.A., Rebholz, L.G.: On conservation laws of Navier–Stokes Galerkin discretizations. J. Comput. Phys. 337, 289–308 (2017)MathSciNet
54.
Zurück zum Zitat Chen, H.: Pointwise error estimates for finite element solutions of the Stokes problem. SIAM J. Numer. Anal. 44(1), 1–28 (2006)MathSciNetMATH Chen, H.: Pointwise error estimates for finite element solutions of the Stokes problem. SIAM J. Numer. Anal. 44(1), 1–28 (2006)MathSciNetMATH
55.
Zurück zum Zitat Chizhonkov, E.V., Olshanskii, M.A.: On the domain geometry dependence of the LBB condition. M2AN Math. Model. Numer. Anal. 34(5), 935–951 (2000)MathSciNetMATH Chizhonkov, E.V., Olshanskii, M.A.: On the domain geometry dependence of the LBB condition. M2AN Math. Model. Numer. Anal. 34(5), 935–951 (2000)MathSciNetMATH
56.
Zurück zum Zitat Codina, R., Blasco, J.: A finite element formulation for the Stokes problem allowing equal velocity–pressure interpolation. Comput. Methods Appl. Mech. Eng. 143(3–4), 373–391 (1997)MathSciNetMATH Codina, R., Blasco, J.: A finite element formulation for the Stokes problem allowing equal velocity–pressure interpolation. Comput. Methods Appl. Mech. Eng. 143(3–4), 373–391 (1997)MathSciNetMATH
57.
Zurück zum Zitat Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7(R–3), 33–75 (1973)MathSciNetMATH Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7(R–3), 33–75 (1973)MathSciNetMATH
58.
Zurück zum Zitat Dallmann, H., Arndt, D.: Stabilized finite element methods for the Oberbeck–Boussinesq model. J. Sci. Comput. 69(1), 244–273 (2016)MathSciNetMATH Dallmann, H., Arndt, D.: Stabilized finite element methods for the Oberbeck–Boussinesq model. J. Sci. Comput. 69(1), 244–273 (2016)MathSciNetMATH
59.
Zurück zum Zitat de Frutos, J., García-Archilla, B., John, V., Novo, J.: An adaptive SUPG method for evolutionary convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 273, 219–237 (2014)MathSciNetMATH de Frutos, J., García-Archilla, B., John, V., Novo, J.: An adaptive SUPG method for evolutionary convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 273, 219–237 (2014)MathSciNetMATH
60.
Zurück zum Zitat de Frutos, J., García-Archilla, B., John, V., Novo, J.: Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements. Adv. Comput. Math. 44, 195–225 (2018)MathSciNetMATH de Frutos, J., García-Archilla, B., John, V., Novo, J.: Analysis of the grad-div stabilization for the time-dependent Navier-Stokes equations with inf-sup stable finite elements. Adv. Comput. Math. 44, 195–225 (2018)MathSciNetMATH
61.
Zurück zum Zitat de Frutos, J., García-Archilla, B., John, V., Novo, J.: Error Analysis of Non Inf-sup Stable Discretizations of the time-dependent Navier–Stokes equations with Local Projection Stabilization. Technical Report arXiv:1709.01011 (2017) de Frutos, J., García-Archilla, B., John, V., Novo, J.: Error Analysis of Non Inf-sup Stable Discretizations of the time-dependent Navier–Stokes equations with Local Projection Stabilization. Technical Report arXiv:​1709.​01011 (2017)
62.
Zurück zum Zitat de Frutos, J., García-Archilla, B., Novo, J.: Local error estimates for the SUPG method applied to evolutionary convection–reaction–diffusion equations. J. Sci. Comput. 66(2), 528–554 (2016)MathSciNetMATH de Frutos, J., García-Archilla, B., Novo, J.: Local error estimates for the SUPG method applied to evolutionary convection–reaction–diffusion equations. J. Sci. Comput. 66(2), 528–554 (2016)MathSciNetMATH
63.
Zurück zum Zitat Dohrmann, C.R., Bochev, P.B.: A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Methods Fluids 46(2), 183–201 (2004)MathSciNetMATH Dohrmann, C.R., Bochev, P.B.: A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int. J. Numer. Methods Fluids 46(2), 183–201 (2004)MathSciNetMATH
64.
Zurück zum Zitat Douglas Jr., J., Wang, J.P.: An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52(186), 495–508 (1989)MathSciNetMATH Douglas Jr., J., Wang, J.P.: An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52(186), 495–508 (1989)MathSciNetMATH
65.
Zurück zum Zitat Du, S., Zhang, Z.: A robust residual-type a posteriori error estimator for convection–diffusion equations. J. Sci. Comput. 65(1), 138–170 (2015)MathSciNetMATH Du, S., Zhang, Z.: A robust residual-type a posteriori error estimator for convection–diffusion equations. J. Sci. Comput. 65(1), 138–170 (2015)MathSciNetMATH
66.
Zurück zum Zitat Durango, F., Novo, J.: Two-grid mixed finite-element approximations to the Navier-Stokes equations based on a Newton type-step. J. Sci. Comput. 74, 456–473 (2018)MathSciNetMATH Durango, F., Novo, J.: Two-grid mixed finite-element approximations to the Navier-Stokes equations based on a Newton type-step. J. Sci. Comput. 74, 456–473 (2018)MathSciNetMATH
67.
Zurück zum Zitat Eigel, M., Merdon, C.: Equilibration a posteriori error estimation for convection–diffusion–reaction problems. J. Sci. Comput. 67(2), 747–768 (2016)MathSciNetMATH Eigel, M., Merdon, C.: Equilibration a posteriori error estimation for convection–diffusion–reaction problems. J. Sci. Comput. 67(2), 747–768 (2016)MathSciNetMATH
68.
Zurück zum Zitat Elman, H., Howle, V.E., Shadid, J., Shuttleworth, R., Tuminaro, R.: Block preconditioners based on approximate commutators. SIAM J. Sci. Comput. 27(5), 1651–1668 (2006)MathSciNetMATH Elman, H., Howle, V.E., Shadid, J., Shuttleworth, R., Tuminaro, R.: Block preconditioners based on approximate commutators. SIAM J. Sci. Comput. 27(5), 1651–1668 (2006)MathSciNetMATH
69.
Zurück zum Zitat Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2014). Numerical Mathematics and Scientific ComputationMATH Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, 2nd edn. Oxford University Press, Oxford (2014). Numerical Mathematics and Scientific ComputationMATH
70.
Zurück zum Zitat Evans, J.A., Hughes, T.J.R.: Isogeometric divergence-conforming B-splines for the steady Navier–Stokes equations. Math. Models Methods Appl. Sci. 23(8), 1421–1478 (2013)MathSciNetMATH Evans, J.A., Hughes, T.J.R.: Isogeometric divergence-conforming B-splines for the steady Navier–Stokes equations. Math. Models Methods Appl. Sci. 23(8), 1421–1478 (2013)MathSciNetMATH
71.
Zurück zum Zitat Evans, J.A., Hughes, T.J.R.: Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations. J. Comput. Phys 241, 141–167 (2013)MathSciNetMATH Evans, J.A., Hughes, T.J.R.: Isogeometric divergence-conforming B-splines for the unsteady Navier–Stokes equations. J. Comput. Phys 241, 141–167 (2013)MathSciNetMATH
72.
Zurück zum Zitat Falk, R.S., Neilan, M.: Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51(2), 1308–1326 (2013)MathSciNetMATH Falk, R.S., Neilan, M.: Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51(2), 1308–1326 (2013)MathSciNetMATH
73.
Zurück zum Zitat Girault, V., Nochetto, R.H., Scott, L.R.: Max-norm estimates for Stokes and Navier–Stokes approximations in convex polyhedra. Numer. Math. 131(4), 771–822 (2015)MathSciNetMATH Girault, V., Nochetto, R.H., Scott, L.R.: Max-norm estimates for Stokes and Navier–Stokes approximations in convex polyhedra. Numer. Math. 131(4), 771–822 (2015)MathSciNetMATH
74.
Zurück zum Zitat Girault, V., Raviart, P.-A.: Finite Element Approximation of the Navier–Stokes Equations, Volume 749 of Lecture Notes in Mathematics. Springer, Berlin (1979) Girault, V., Raviart, P.-A.: Finite Element Approximation of the Navier–Stokes Equations, Volume 749 of Lecture Notes in Mathematics. Springer, Berlin (1979)
75.
Zurück zum Zitat Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. Theory and algorithms. In: Volume 5 of Springer Series in Computational Mathematics. Springer, Berlin (1986) Girault, V., Raviart, P.-A.: Finite element methods for Navier-Stokes equations. Theory and algorithms. In: Volume 5 of Springer Series in Computational Mathematics. Springer, Berlin (1986)
76.
Zurück zum Zitat Girault, V., Scott, L.R.: A quasi-local interpolation operator preserving the discrete divergence. Calcolo 40(1), 1–19 (2003)MathSciNetMATH Girault, V., Scott, L.R.: A quasi-local interpolation operator preserving the discrete divergence. Calcolo 40(1), 1–19 (2003)MathSciNetMATH
77.
Zurück zum Zitat Glowinski, R.: Finite element methods for incompressible viscous flow. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. IX, pp. 3–1176. North-Holland, Amsterdam (2003) Glowinski, R.: Finite element methods for incompressible viscous flow. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. IX, pp. 3–1176. North-Holland, Amsterdam (2003)
78.
Zurück zum Zitat Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.) 47(89), 271–306 (1959)MathSciNetMATH Godunov, S.K.: A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.) 47(89), 271–306 (1959)MathSciNetMATH
79.
Zurück zum Zitat Guzmán, J., Leykekhman, D.: Pointwise error estimates of finite element approximations to the Stokes problem on convex polyhedra. Math. Comput. 81(280), 1879–1902 (2012)MathSciNetMATH Guzmán, J., Leykekhman, D.: Pointwise error estimates of finite element approximations to the Stokes problem on convex polyhedra. Math. Comput. 81(280), 1879–1902 (2012)MathSciNetMATH
80.
Zurück zum Zitat Guzmán, J., Neilan, M.: Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83(285), 15–36 (2014)MathSciNetMATH Guzmán, J., Neilan, M.: Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83(285), 15–36 (2014)MathSciNetMATH
81.
Zurück zum Zitat Guzmán, J., Sánchez, M.A.: Max-norm stability of low order Taylor–Hood elements in three dimensions. J. Sci. Comput. 65(2), 598–621 (2015)MathSciNetMATH Guzmán, J., Sánchez, M.A.: Max-norm stability of low order Taylor–Hood elements in three dimensions. J. Sci. Comput. 65(2), 598–621 (2015)MathSciNetMATH
82.
Zurück zum Zitat Hauke, G., Doweidar, M.H., Fuster, D.: A posteriori error estimation for computational fluid dynamics: the variational multiscale approach. In: de Borst R., Ramm E. (eds) Multiscale Methods in Computational Mechanics, Lecture Notes in Applied and Computational Mechanics, vol. 55. Springer, Dordrecht (2010) Hauke, G., Doweidar, M.H., Fuster, D.: A posteriori error estimation for computational fluid dynamics: the variational multiscale approach. In: de Borst R., Ramm E. (eds) Multiscale Methods in Computational Mechanics, Lecture Notes in Applied and Computational Mechanics, vol. 55. Springer, Dordrecht (2010)
83.
Zurück zum Zitat Hauke, G., Doweidar, M.H., Fuster, D., Gómez, A., Sayas, J.: Application of variational a-posteriori multiscale error estimation to higher-order elements. Comput. Mech. 38(4–5), 356–389 (2006)MathSciNetMATH Hauke, G., Doweidar, M.H., Fuster, D., Gómez, A., Sayas, J.: Application of variational a-posteriori multiscale error estimation to higher-order elements. Comput. Mech. 38(4–5), 356–389 (2006)MathSciNetMATH
84.
Zurück zum Zitat Hauke, G., Fuster, D., Doweidar, M.H.: Variational multiscale a-posteriori error estimation for multi-dimensional transport problems. Comput. Methods Appl. Mech. Eng. 197(33–40), 2701–2718 (2008)MathSciNetMATH Hauke, G., Fuster, D., Doweidar, M.H.: Variational multiscale a-posteriori error estimation for multi-dimensional transport problems. Comput. Methods Appl. Mech. Eng. 197(33–40), 2701–2718 (2008)MathSciNetMATH
85.
Zurück zum Zitat Hosseini, B.S., Möller, M., Turek, S.: Isogeometric analysis of the Navier–Stokes equations with Taylor–Hood B-spline elements. Appl. Math. Comput. 267, 264–281 (2015)MathSciNetMATH Hosseini, B.S., Möller, M., Turek, S.: Isogeometric analysis of the Navier–Stokes equations with Taylor–Hood B-spline elements. Appl. Math. Comput. 267, 264–281 (2015)MathSciNetMATH
86.
Zurück zum Zitat Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)MathSciNetMATH Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)MathSciNetMATH
87.
Zurück zum Zitat Hughes, T.J.R.: Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng. 127(1–4), 387–401 (1995)MathSciNetMATH Hughes, T.J.R.: Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng. 127(1–4), 387–401 (1995)MathSciNetMATH
88.
Zurück zum Zitat 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), Volume 34 of AMD, pp. 19–35. Amer. Soc. Mech. Engrs. (ASME), New York (1979) 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), Volume 34 of AMD, pp. 19–35. Amer. Soc. Mech. Engrs. (ASME), New York (1979)
89.
Zurück zum Zitat Hughes, T.J.R., Franca, L.P.: A new finite element formulation for computational fluid dynamics. VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces. Comput. Methods Appl. Mech. Eng. 65(1), 85–96 (1987)MathSciNetMATH Hughes, T.J.R., Franca, L.P.: A new finite element formulation for computational fluid dynamics. VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces. Comput. Methods Appl. Mech. Eng. 65(1), 85–96 (1987)MathSciNetMATH
90.
Zurück zum Zitat Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška–Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59(1), 85–99 (1986)MATH Hughes, T.J.R., Franca, L.P., Balestra, M.: A new finite element formulation for computational fluid dynamics. V. Circumventing the Babuška–Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Eng. 59(1), 85–99 (1986)MATH
91.
Zurück zum Zitat Hughes, T.J.R., Sangalli, G.: Variational multiscale analysis: the fine-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal. 45(2), 539–557 (2007)MathSciNetMATH Hughes, T.J.R., Sangalli, G.: Variational multiscale analysis: the fine-scale Green’s function, projection, optimization, localization, and stabilized methods. SIAM J. Numer. Anal. 45(2), 539–557 (2007)MathSciNetMATH
92.
Zurück zum Zitat John, V.: A numerical study of a posteriori error estimators for convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 190(5–7), 757–781 (2000)MathSciNetMATH John, V.: A numerical study of a posteriori error estimators for convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 190(5–7), 757–781 (2000)MathSciNetMATH
93.
Zurück zum Zitat John, V.: Finite element methods for incompressible flow problems, vol. 51 of Springer Series in Computational Mathematics. Springer, Cham (2016) John, V.: Finite element methods for incompressible flow problems, vol. 51 of Springer Series in Computational Mathematics. Springer, Cham (2016)
94.
Zurück zum Zitat John, V., Kaiser, K., Novo, J.: Finite element methods for the incompressible Stokes equations with variable viscosity. ZAMM Z. Angew. Math. Mech. 96(2), 205–216 (2016)MathSciNet John, V., Kaiser, K., Novo, J.: Finite element methods for the incompressible Stokes equations with variable viscosity. ZAMM Z. Angew. Math. Mech. 96(2), 205–216 (2016)MathSciNet
95.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
96.
Zurück zum Zitat John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. II. Analysis for \(P_1\) and \(Q_1\) finite elements. Comput. Methods Appl. Mech. Eng. 197(21–24), 1997–2014 (2008)MATH John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. II. Analysis for \(P_1\) and \(Q_1\) finite elements. Comput. Methods Appl. Mech. Eng. 197(21–24), 1997–2014 (2008)MATH
97.
Zurück zum Zitat John, V., Layton, W., Manica, C.C.: Convergence of time-averaged statistics of finite element approximations of the Navier–Stokes equations. SIAM J. Numer. Anal. 46(1), 151–179 (2007)MathSciNetMATH John, V., Layton, W., Manica, C.C.: Convergence of time-averaged statistics of finite element approximations of the Navier–Stokes equations. SIAM J. Numer. Anal. 46(1), 151–179 (2007)MathSciNetMATH
98.
Zurück zum Zitat John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59, 492–544 (2017)MathSciNetMATH John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59, 492–544 (2017)MathSciNetMATH
99.
Zurück zum Zitat John, V., Mitkova, T., Roland, M., Sundmacher, K., Tobiska, L., Voigt, A.: Simulations of population balance systems with one internal coordinate using finite element methods. Chem. Eng. Sci. 64(4), 733–741 (2009) John, V., Mitkova, T., Roland, M., Sundmacher, K., Tobiska, L., Voigt, A.: Simulations of population balance systems with one internal coordinate using finite element methods. Chem. Eng. Sci. 64(4), 733–741 (2009)
100.
Zurück zum Zitat John, V., Novo, J.: On (essentially) non-oscillatory discretizations of evolutionary convection–diffusion equations. J. Comput. Phys. 231(4), 1570–1586 (2012)MathSciNetMATH John, V., Novo, J.: On (essentially) non-oscillatory discretizations of evolutionary convection–diffusion equations. J. Comput. Phys. 231(4), 1570–1586 (2012)MathSciNetMATH
101.
Zurück zum Zitat John, V., Novo, J.: A robust SUPG norm a posteriori error estimator for stationary convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 255, 289–305 (2013)MathSciNetMATH John, V., Novo, J.: A robust SUPG norm a posteriori error estimator for stationary convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 255, 289–305 (2013)MathSciNetMATH
102.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
103.
Zurück zum Zitat John, V., Schumacher, L.: A study of isogeometric analysis for scalar convection–diffusion equations. Appl. Math. Lett. 27, 43–48 (2014)MathSciNetMATH John, V., Schumacher, L.: A study of isogeometric analysis for scalar convection–diffusion equations. Appl. Math. Lett. 27, 43–48 (2014)MathSciNetMATH
104.
Zurück zum Zitat Johnson, C., Schatz, A.H., Wahlbin, L.B.: Crosswind smear and pointwise errors in streamline diffusion finite element methods. Math. Comput. 49(179), 25–38 (1987)MathSciNetMATH Johnson, C., Schatz, A.H., Wahlbin, L.B.: Crosswind smear and pointwise errors in streamline diffusion finite element methods. Math. Comput. 49(179), 25–38 (1987)MathSciNetMATH
105.
Zurück zum Zitat Knobloch, P.: Improvements of the Mizukami–Hughes method for convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 196(1–3), 579–594 (2006)MathSciNetMATH Knobloch, P.: Improvements of the Mizukami–Hughes method for convection–diffusion equations. Comput. Methods Appl. Mech. Eng. 196(1–3), 579–594 (2006)MathSciNetMATH
106.
Zurück zum Zitat Knopp, T., Lube, G., Rapin, G.: Stabilized finite element methods with shock capturing for advection–diffusion problems. Comput. Methods Appl. Mech. Eng. 191(27–28), 2997–3013 (2002)MathSciNetMATH Knopp, T., Lube, G., Rapin, G.: Stabilized finite element methods with shock capturing for advection–diffusion problems. Comput. Methods Appl. Mech. Eng. 191(27–28), 2997–3013 (2002)MathSciNetMATH
107.
Zurück zum Zitat 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)MathSciNetMATH 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)MathSciNetMATH
108.
Zurück zum Zitat Kuzmin, D.: Algebraic flux correction for finite element discretizations of coupled systems. In: Manolis, P., Eugenio, O., Bernard, S. (eds.) Proceedings of the International Conference on Computational Methods for Coupled Problems in Science and Engineering, pp. 1–5. CIMNE, Barcelona (2007) Kuzmin, D.: Algebraic flux correction for finite element discretizations of coupled systems. In: Manolis, P., Eugenio, O., Bernard, S. (eds.) Proceedings of the International Conference on Computational Methods for Coupled Problems in Science and Engineering, pp. 1–5. CIMNE, Barcelona (2007)
109.
Zurück zum Zitat Kuzmin, D.: Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes. J. Comput. Appl. Math. 236(9), 2317–2337 (2012)MathSciNetMATH Kuzmin, D.: Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes. J. Comput. Appl. Math. 236(9), 2317–2337 (2012)MathSciNetMATH
110.
Zurück zum Zitat 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., 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)
111.
Zurück zum Zitat Kuzmin, D., Turek, S.: High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. J. Comput. Phys. 198(1), 131–158 (2004)MathSciNetMATH Kuzmin, D., Turek, S.: High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. J. Comput. Phys. 198(1), 131–158 (2004)MathSciNetMATH
112.
Zurück zum Zitat Layton, W.: Introduction to the Numerical Analysis of Incompressible Viscous Flows, Volume 6 of Computational Science & Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008) Layton, W.: Introduction to the Numerical Analysis of Incompressible Viscous Flows, Volume 6 of Computational Science & Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)
113.
Zurück zum Zitat Lederer, P.L., Linke, A., Merdon, C., Schöberl, J.: Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements. SIAM J. Numer. Anal. 55(3), 1291–1314 (2017)MathSciNetMATH Lederer, P.L., Linke, A., Merdon, C., Schöberl, J.: Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements. SIAM J. Numer. Anal. 55(3), 1291–1314 (2017)MathSciNetMATH
114.
Zurück zum Zitat Liao, Q., Silvester, D.: Robust stabilized Stokes approximation methods for highly stretched grids. IMA J. Numer. Anal. 33(2), 413–431 (2013)MathSciNetMATH Liao, Q., Silvester, D.: Robust stabilized Stokes approximation methods for highly stretched grids. IMA J. Numer. Anal. 33(2), 413–431 (2013)MathSciNetMATH
115.
Zurück zum Zitat Linke, A.: On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Eng. 268, 782–800 (2014)MathSciNetMATH Linke, A.: On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime. Comput. Methods Appl. Mech. Eng. 268, 782–800 (2014)MathSciNetMATH
116.
Zurück zum Zitat Linke, A., Matthies, G., Tobiska, L.: Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors. ESAIM Math. Model. Numer. Anal. 50(1), 289–309 (2016)MathSciNetMATH Linke, A., Matthies, G., Tobiska, L.: Robust arbitrary order mixed finite element methods for the incompressible Stokes equations with pressure independent velocity errors. ESAIM Math. Model. Numer. Anal. 50(1), 289–309 (2016)MathSciNetMATH
117.
Zurück zum Zitat Linke, A., Merdon, C.: Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 311, 304–326 (2016)MathSciNet Linke, A., Merdon, C.: Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 311, 304–326 (2016)MathSciNet
118.
Zurück zum Zitat 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)MATH 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)MATH
119.
Zurück zum Zitat Lube, G., Arndt, D., Dallmann, H.: Understanding the limits of inf-sup stable Galerkin-FEM for incompressible flows. In: Boundary and interior layers, computational and asymptotic methods—BAIL 2014, volume 108 of Lect. Notes Comput. Sci. Eng., pp. 147–169. Springer, Cham (2015) Lube, G., Arndt, D., Dallmann, H.: Understanding the limits of inf-sup stable Galerkin-FEM for incompressible flows. In: Boundary and interior layers, computational and asymptotic methods—BAIL 2014, volume 108 of Lect. Notes Comput. Sci. Eng., pp. 147–169. Springer, Cham (2015)
120.
Zurück zum Zitat Lube, G., Rapin, G.: Residual-based stabilized higher-order FEM for advection-dominated problems. Comput. Methods Appl. Mech. Eng. 195(33–36), 4124–4138 (2006)MathSciNetMATH Lube, G., Rapin, G.: Residual-based stabilized higher-order FEM for advection-dominated problems. Comput. Methods Appl. Mech. Eng. 195(33–36), 4124–4138 (2006)MathSciNetMATH
121.
Zurück zum Zitat Micheletti, S., Perotto, S., Picasso, M.: Stabilized finite elements on anisotropic meshes: a priori error estimates for the advection–diffusion and the Stokes problems. SIAM J. Numer. Anal. 41(3), 1131–1162 (2003)MathSciNetMATH Micheletti, S., Perotto, S., Picasso, M.: Stabilized finite elements on anisotropic meshes: a priori error estimates for the advection–diffusion and the Stokes problems. SIAM J. Numer. Anal. 41(3), 1131–1162 (2003)MathSciNetMATH
122.
Zurück zum Zitat Mizukami, A., Hughes, T.J.R.: A Petrov–Galerkin finite element method for convection-dominated flows: an accurate upwinding technique for satisfying the maximum principle. Comput. Methods Appl. Mech. Eng. 50(2), 181–193 (1985)MathSciNetMATH Mizukami, A., Hughes, T.J.R.: A Petrov–Galerkin finite element method for convection-dominated flows: an accurate upwinding technique for satisfying the maximum principle. Comput. Methods Appl. Mech. Eng. 50(2), 181–193 (1985)MathSciNetMATH
123.
Zurück zum Zitat Nävert, U.: A finite element method for convection–diffusion problems. Ph.D. Thesis, Chalmers University of Technology (1982) Nävert, U.: A finite element method for convection–diffusion problems. Ph.D. Thesis, Chalmers University of Technology (1982)
124.
Zurück zum Zitat Niijima, K.: Pointwise error estimates for a streamline diffusion finite element scheme. Numer. Math. 56(7), 707–719 (1990)MathSciNetMATH Niijima, K.: Pointwise error estimates for a streamline diffusion finite element scheme. Numer. Math. 56(7), 707–719 (1990)MathSciNetMATH
125.
Zurück zum Zitat Roos, H.-G., Stynes, M.: Some open questions in the numerical analysis of singularly perturbed differential equations. Comput. Methods Appl. Math. 15(4), 531–550 (2015)MathSciNetMATH Roos, H.-G., Stynes, M.: Some open questions in the numerical analysis of singularly perturbed differential equations. Comput. Methods Appl. Math. 15(4), 531–550 (2015)MathSciNetMATH
126.
Zurück zum Zitat Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Convection–Diffusion and Flow Problems, vol. 24 of Springer Series in Computational Mathematics. Springer, Berlin (1996) Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Convection–Diffusion and Flow Problems, vol. 24 of Springer Series in Computational Mathematics. Springer, Berlin (1996)
127.
Zurück zum Zitat Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems, vol. 24 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2008) Roos, H.-G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion-Reaction and Flow Problems, vol. 24 of Springer Series in Computational Mathematics, 2nd edn. Springer, Berlin (2008)
128.
Zurück zum Zitat Saad, Y.: A flexible inner–outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993)MathSciNetMATH Saad, Y.: A flexible inner–outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993)MathSciNetMATH
129.
Zurück zum Zitat Sangalli, G.: Robust a-posteriori estimator for advection–diffusion–reaction problems. Math. Comput. 77(261), 41–70 (2008). (electronic)MathSciNetMATH Sangalli, G.: Robust a-posteriori estimator for advection–diffusion–reaction problems. Math. Comput. 77(261), 41–70 (2008). (electronic)MathSciNetMATH
130.
Zurück zum Zitat Schötzau, D., Schwab, C., Stenberg, R.: Mixed \(hp\)-FEM on anisotropic meshes. II. Hanging nodes and tensor products of boundary layer meshes. Numer. Math. 83(4), 667–697 (1999)MathSciNetMATH Schötzau, D., Schwab, C., Stenberg, R.: Mixed \(hp\)-FEM on anisotropic meshes. II. Hanging nodes and tensor products of boundary layer meshes. Numer. Math. 83(4), 667–697 (1999)MathSciNetMATH
131.
Zurück zum Zitat Schroeder, P.W., Lube, G.: Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier-Stokes flows. J. Num. Math., Accepted for publication (2017) Schroeder, P.W., Lube, G.: Pressure-robust analysis of divergence-free and conforming FEM for evolutionary incompressible Navier-Stokes flows. J. Num. Math., Accepted for publication (2017)
132.
Zurück zum Zitat Schwegler, K., Bause, M.: Goal-oriented a posteriori error control for nonstationary convection-dominated transport problems. Technical Report arXiv:1601.06544 (2016) Schwegler, K., Bause, M.: Goal-oriented a posteriori error control for nonstationary convection-dominated transport problems. Technical Report arXiv:​1601.​06544 (2016)
133.
Zurück zum Zitat Scott, L.R., Vogelius, M.: Conforming finite element methods for incompressible and nearly incompressible continua. In: Large-Scale Computations in Fluid Mechanics, Part 2 (La Jolla, Calif., 1983), Volume 22 of Lectures in Appl. Math., pp. 221–244. Amer. Math. Soc., Providence (1985) Scott, L.R., Vogelius, M.: Conforming finite element methods for incompressible and nearly incompressible continua. In: Large-Scale Computations in Fluid Mechanics, Part 2 (La Jolla, Calif., 1983), Volume 22 of Lectures in Appl. Math., pp. 221–244. Amer. Math. Soc., Providence (1985)
134.
Zurück zum Zitat Speleers, H., Manni, C., Pelosi, F., Sampoli, M.L.: Isogeometric analysis with Powell–Sabin splines for advection–diffusion–reaction problems. Comput. Methods Appl. Mech. Eng. 221/222, 132–148 (2012)MathSciNetMATH Speleers, H., Manni, C., Pelosi, F., Sampoli, M.L.: Isogeometric analysis with Powell–Sabin splines for advection–diffusion–reaction problems. Comput. Methods Appl. Mech. Eng. 221/222, 132–148 (2012)MathSciNetMATH
135.
Zurück zum Zitat Tabata, M., Tagami, D.: Error estimates for finite element approximations of drag and lift in nonstationary Navier–Stokes flows. Japan J. Ind. Appl. Math. 17(3), 371–389 (2000)MathSciNetMATH Tabata, M., Tagami, D.: Error estimates for finite element approximations of drag and lift in nonstationary Navier–Stokes flows. Japan J. Ind. Appl. Math. 17(3), 371–389 (2000)MathSciNetMATH
136.
Zurück zum Zitat Tobiska, L., Verfürth, R.: Robust a posteriori error estimates for stabilized finite element methods. IMA J. Numer. Anal. 35(4), 1652–1671 (2015)MathSciNetMATH Tobiska, L., Verfürth, R.: Robust a posteriori error estimates for stabilized finite element methods. IMA J. Numer. Anal. 35(4), 1652–1671 (2015)MathSciNetMATH
137.
Zurück zum Zitat Vanka, S.P.: Block-implicit multigrid solution of Navier–Stokes equations in primitive variables. J. Comput. Phys. 65(1), 138–158 (1986)MathSciNetMATH Vanka, S.P.: Block-implicit multigrid solution of Navier–Stokes equations in primitive variables. J. Comput. Phys. 65(1), 138–158 (1986)MathSciNetMATH
138.
Zurück zum Zitat Verfürth, R.: A posteriori error estimators for convection–diffusion equations. Numer. Math. 80(4), 641–663 (1998)MathSciNetMATH Verfürth, R.: A posteriori error estimators for convection–diffusion equations. Numer. Math. 80(4), 641–663 (1998)MathSciNetMATH
139.
Zurück zum Zitat Verfürth, R.: Robust a posteriori error estimates for stationary convection–diffusion equations. SIAM J. Numer. Anal. 43(4), 1766–1782 (2005). (electronic)MathSciNetMATH Verfürth, R.: Robust a posteriori error estimates for stationary convection–diffusion equations. SIAM J. Numer. Anal. 43(4), 1766–1782 (2005). (electronic)MathSciNetMATH
140.
Zurück zum Zitat Wilbrandt, U., Bartsch, C., Ahmed, N., Alia, N., Anker, F., Blank, L., Caiazzo, A., Ganesan, S., Giere, S., Matthies, G., Meesala, R., Shamim, A., Venkatesan, J., John, V.: ParMooN—a modernized program package based on mapped finite elements. Comput. Math. Appl. 74(1), 74–88 (2017)MathSciNetMATH Wilbrandt, U., Bartsch, C., Ahmed, N., Alia, N., Anker, F., Blank, L., Caiazzo, A., Ganesan, S., Giere, S., Matthies, G., Meesala, R., Shamim, A., Venkatesan, J., John, V.: ParMooN—a modernized program package based on mapped finite elements. Comput. Math. Appl. 74(1), 74–88 (2017)MathSciNetMATH
141.
Zurück zum Zitat Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31(3), 335–362 (1979)MathSciNetMATH Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31(3), 335–362 (1979)MathSciNetMATH
142.
Zurück zum Zitat Zhang, S.: A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74(250), 543–554 (2005)MathSciNetMATH Zhang, S.: A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74(250), 543–554 (2005)MathSciNetMATH
143.
Zurück zum Zitat Zhou, G.H., Rannacher, R.: Pointwise superconvergence of the streamline diffusion finite-element method. Numer. Methods Partial Differ. Equ 12(1), 123–145 (1996)MathSciNetMATH Zhou, G.H., Rannacher, R.: Pointwise superconvergence of the streamline diffusion finite-element method. Numer. Methods Partial Differ. Equ 12(1), 123–145 (1996)MathSciNetMATH
144.
Zurück zum Zitat Zhou, G.: How accurate is the streamline diffusion finite element method? Math. Comput. 66(217), 31–44 (1997)MathSciNetMATH Zhou, G.: How accurate is the streamline diffusion finite element method? Math. Comput. 66(217), 31–44 (1997)MathSciNetMATH
Metadaten
Titel
Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story?
verfasst von
Volker John
Petr Knobloch
Julia Novo
Publikationsdatum
05.02.2018
Verlag
Springer Berlin Heidelberg
Erschienen in
Computing and Visualization in Science / Ausgabe 5-6/2018
Print ISSN: 1432-9360
Elektronische ISSN: 1433-0369
DOI
https://doi.org/10.1007/s00791-018-0290-5

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