Finite elements for scalar convection-dominated equations and incompressible flow problems: a never ending story? (English) Zbl 07704544
Summary: 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.
MSC:
| 65Nxx | Numerical methods for partial differential equations, boundary value problems |
References:
| [1] | 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) · Zbl 0948.65115 |
| [2] | 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) · Zbl 1439.76040 |
| [3] | 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) · Zbl 1322.76038 |
| [4] | Ainsworth, M., Coggins, P.: The stability of mixed \[hp\] hp-finite element methods for Stokes flow on high aspect ratio elements. SIAM J. Numer. Anal. 38(5), 1721-1761 (2000) · Zbl 0989.76039 |
| [5] | 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) · Zbl 1433.76065 |
| [6] | 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) · Zbl 1148.76029 |
| [7] | Apel, T., Randrianarivony, H.M.: Stability of discretizations of the Stokes problem on anisotropic meshes. Math. Comput. Simul. 61(3-6), 437-447 (2003) · Zbl 1205.76072 |
| [8] | 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) · Zbl 1205.65307 |
| [9] | Apel, T., Nicaise, S.: The inf-sup condition for low order elements on anisotropic meshes. Calcolo 41(2), 89-113 (2004) · Zbl 1108.65117 |
| [10] | 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) · Zbl 0998.65116 |
| [11] | Arminjon, P., Dervieux, A.: Construction of TVD-like artificial viscosities on two-dimensional arbitrary FEM grids. J. Comput. Phys. 106(1), 176-198 (1993) · Zbl 0771.65062 |
| [12] | 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) · Zbl 1446.76126 |
| [13] | 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) · Zbl 1230.76021 |
| [14] | Babuška, I.: Error-bounds for finite element method. Numer. Math. 16, 322-333 (1971) · Zbl 0214.42001 |
| [15] | Bardos, C.W., Titi, E.S.: Mathematics and turbulence: where do we stand? J. Turbul. 14(3), 42-76 (2013) |
| [16] | 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) · Zbl 1303.65082 |
| [17] | 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) · Zbl 1332.65105 |
| [18] | Barrenechea, G.R., John, V., Knobloch, P.: Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal. 54(4), 2427-2451 (2016) · Zbl 1346.65048 |
| [19] | 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) · Zbl 1360.65275 |
| [20] | Barrenechea, G.R., Valentin, F.: Consistent local projection stabilized finite element methods. SIAM J. Numer. Anal. 48(5), 1801-1825 (2010) · Zbl 1219.65131 |
| [21] | 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) · Zbl 1231.76135 |
| [22] | 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) · Zbl 1235.76059 |
| [23] | 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) · Zbl 1439.76049 |
| [24] | Bazilevs, Y., Beirão da Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for \[h\] h-refined meshes. Math. Models Methods Appl. Sci. 16(7), 1031-1090 (2006) · Zbl 1103.65113 |
| [25] | Bazilevs, Y., Calo, V.M., Tezduyar, T.E., Hughes, T.J.R.: \[YZ\beta\] YZβ discontinuity capturing for advection-dominated processes with application to arterial drug delivery. Int. J. Numer. Methods Fluids 54(6-8), 593-608 (2007) · Zbl 1207.76049 |
| [26] | Becker, R., Braack, M.: A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38(4), 173-199 (2001) · Zbl 1008.76036 |
| [27] | Becker, R.; Braack, M.; Feistauer, M. (ed.); Dolejší, V. (ed.); Knobloch, P. (ed.); Najzar, K. (ed.), A two-level stabilization scheme for the Navier-Stokes equations, 123-130 (2004), Berlin · Zbl 1198.76062 |
| [28] | Benzi, M., Olshanskii, M.A.: An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput. 28(6), 2095-2113 (2006) · Zbl 1126.76028 |
| [29] | 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) · Zbl 1298.76114 |
| [30] | Berrone, S.: Robustness in a posteriori error analysis for FEM flow models. Numer. Math. 91(3), 389-422 (2002) · Zbl 1009.76049 |
| [31] | 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) · Zbl 1159.76346 |
| [32] | 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) · Zbl 0251.76004 |
| [33] | 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) · Zbl 1149.76026 |
| [34] | 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) · Zbl 1298.65146 |
| [35] | Braack, M., Lube, G., Röhe, L.: Divergence preserving interpolation on anisotropic quadrilateral meshes. Comput. Methods Appl. Math. 12(2), 123-138 (2012) · Zbl 1284.41001 |
| [36] | Braack, M., Mucha, P.B.: Directional do-nothing condition for the Navier-Stokes equations. J. Comput. Math. 32(5), 507-521 (2014) · Zbl 1324.76015 |
| [37] | Brennecke, C., Linke, A., Merdon, C., Schöberl, J.: Optimal and pressure-independent \[L^2\] L2 velocity error estimates for a modified Crouzeix-Raviart Stokes element with BDM reconstructions. J. Comput. Math. 33(2), 191-208 (2015) · Zbl 1340.76024 |
| [38] | 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) · Zbl 0338.90047 |
| [39] | Brezzi, F., Fortin, M.: A minimal stabilisation procedure for mixed finite element methods. Numer. Math. 89(3), 457-491 (2001) · Zbl 1009.65067 |
| [40] | 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) · Zbl 0552.76002 |
| [41] | 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) · Zbl 0497.76041 |
| [42] | 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) · Zbl 1429.76044 |
| [43] | Bulling, J., John, V., Knobloch, P.: Isogeometric analysis for flows around a cylinder. Appl. Math. Lett. 63, 65-70 (2017) · Zbl 1346.76025 |
| [44] | 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) · Zbl 1205.65249 |
| [45] | 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) · Zbl 1423.76214 |
| [46] | 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) · Zbl 1078.65088 |
| [47] | 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) · Zbl 1398.76097 |
| [48] | 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) · Zbl 1117.76032 |
| [49] | 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) · Zbl 1166.65054 |
| [50] | 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) · Zbl 1085.76033 |
| [51] | 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) · Zbl 1125.76038 |
| [52] | 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) · Zbl 1457.76097 |
| [53] | 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) · Zbl 1415.65222 |
| [54] | Chen, H.: Pointwise error estimates for finite element solutions of the Stokes problem. SIAM J. Numer. Anal. 44(1), 1-28 (2006) · Zbl 1112.65108 |
| [55] | 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) · Zbl 1006.76052 |
| [56] | 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) · Zbl 0893.76040 |
| [57] | 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) · Zbl 0302.65087 |
| [58] | Dallmann, H., Arndt, D.: Stabilized finite element methods for the Oberbeck-Boussinesq model. J. Sci. Comput. 69(1), 244-273 (2016) · Zbl 1457.65074 |
| [59] | 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) · Zbl 1296.76090 |
| [60] | 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) · Zbl 1404.65188 |
| [61] | 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) · Zbl 1496.65160 |
| [62] | 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) · Zbl 1336.65151 |
| [63] | 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) · Zbl 1060.76569 |
| [64] | Douglas Jr., J., Wang, J.P.: An absolutely stabilized finite element method for the Stokes problem. Math. Comput. 52(186), 495-508 (1989) · Zbl 0669.76051 |
| [65] | Du, S., Zhang, Z.: A robust residual-type a posteriori error estimator for convection – diffusion equations. J. Sci. Comput. 65(1), 138-170 (2015) · Zbl 1333.65121 |
| [66] | 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) · Zbl 1404.65166 |
| [67] | Eigel, M., Merdon, C.: Equilibration a posteriori error estimation for convection – diffusion – reaction problems. J. Sci. Comput. 67(2), 747-768 (2016) · Zbl 1353.65113 |
| [68] | 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) · Zbl 1100.65042 |
| [69] | 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 Computation · Zbl 1304.76002 |
| [70] | 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) · Zbl 1383.76337 |
| [71] | 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) · Zbl 1349.76054 |
| [72] | 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) · Zbl 1268.76032 |
| [73] | 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) · Zbl 1401.76087 |
| [74] | Girault, V., Raviart, P.-A.: Finite Element Approximation of the Navier-Stokes Equations, Volume 749 of Lecture Notes in Mathematics. Springer, Berlin (1979) · Zbl 0413.65081 |
| [75] | 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) · Zbl 0585.65077 |
| [76] | Girault, V., Scott, L.R.: A quasi-local interpolation operator preserving the discrete divergence. Calcolo 40(1), 1-19 (2003) · Zbl 1072.65014 |
| [77] | Glowinski, R.; Ciarlet, PG (ed.); Lions, JL (ed.), Finite element methods for incompressible viscous flow, No. IX, 3-1176 (2003), Amsterdam · Zbl 1040.76001 |
| [78] | 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) · Zbl 0171.46204 |
| [79] | 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) · Zbl 1307.65165 |
| [80] | Guzmán, J., Neilan, M.: Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83(285), 15-36 (2014) · Zbl 1322.76041 |
| [81] | 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) · Zbl 1327.65240 |
| [82] | 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) · Zbl 1323.76084 |
| [83] | 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) · Zbl 1160.76026 |
| [84] | 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) · Zbl 1194.76119 |
| [85] | 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) · Zbl 1410.76043 |
| [86] | 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) · Zbl 1151.74419 |
| [87] | 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) · Zbl 0866.76044 |
| [88] | 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) · Zbl 0423.76067 |
| [89] | 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) · Zbl 0635.76067 |
| [90] | 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) · Zbl 0622.76077 |
| [91] | 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) · Zbl 1152.65111 |
| [92] | 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) · Zbl 0973.76049 |
| [93] | John, V.: Finite element methods for incompressible flow problems, vol. 51 of Springer Series in Computational Mathematics. Springer, Cham (2016) · Zbl 1358.76003 |
| [94] | 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) · Zbl 1538.65524 |
| [95] | 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) · Zbl 1173.76342 |
| [96] | John, V., Knobloch, P.: On spurious oscillations at layers diminishing (SOLD) methods for convection-diffusion equations. II. Analysis for \[P_1\] P1 and \[Q_1\] Q1 finite elements. Comput. Methods Appl. Mech. Eng. 197(21-24), 1997-2014 (2008) · Zbl 1194.76122 |
| [97] | 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) · Zbl 1157.74029 |
| [98] | 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) · Zbl 1426.76275 |
| [99] | 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] | John, V., Novo, J.: On (essentially) non-oscillatory discretizations of evolutionary convection – diffusion equations. J. Comput. Phys. 231(4), 1570-1586 (2012) · Zbl 1242.65164 |
| [101] | 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) · Zbl 1297.65157 |
| [102] | 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) · Zbl 1228.76088 |
| [103] | John, V., Schumacher, L.: A study of isogeometric analysis for scalar convection – diffusion equations. Appl. Math. Lett. 27, 43-48 (2014) · Zbl 1311.65146 |
| [104] | 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) · Zbl 0629.65111 |
| [105] | Knobloch, P.: Improvements of the Mizukami-Hughes method for convection – diffusion equations. Comput. Methods Appl. Mech. Eng. 196(1-3), 579-594 (2006) · Zbl 1120.76328 |
| [106] | 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) · Zbl 1001.76058 |
| [107] | 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) · Zbl 1189.76342 |
| [108] | Kuzmin, D.; Manolis, P. (ed.); Eugenio, O. (ed.); Bernard, S. (ed.), Algebraic flux correction for finite element discretizations of coupled systems, 1-5 (2007), Barcelona |
| [109] | Kuzmin, D.: Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes. J. Comput. Appl. Math. 236(9), 2317-2337 (2012) · Zbl 1241.65083 |
| [110] | Kuzmin, D.; Möller, M.; Kuzmin, D. (ed.); Löhner, R. (ed.); Turek, S. (ed.), Algebraic flux correction I. Scalar conservation laws, 155-206 (2005), Berlin · Zbl 1094.76040 |
| [111] | Kuzmin, D., Turek, S.: High-resolution FEM-TVD schemes based on a fully multidimensional flux limiter. J. Comput. Phys. 198(1), 131-158 (2004) · Zbl 1107.76352 |
| [112] | 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) · Zbl 1153.76002 |
| [113] | 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) · Zbl 1457.65202 |
| [114] | Liao, Q., Silvester, D.: Robust stabilized Stokes approximation methods for highly stretched grids. IMA J. Numer. Anal. 33(2), 413-431 (2013) · Zbl 1328.76043 |
| [115] | 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) · Zbl 1295.76007 |
| [116] | 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) · Zbl 1381.76186 |
| [117] | 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) · Zbl 1439.76083 |
| [118] | 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) · Zbl 0633.76070 |
| [119] | 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) · Zbl 1427.76136 |
| [120] | 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) · Zbl 1125.76042 |
| [121] | 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) · Zbl 1053.65089 |
| [122] | 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) · Zbl 0553.76075 |
| [123] | Nävert, U.: A finite element method for convection – diffusion problems. Ph.D. Thesis, Chalmers University of Technology (1982) |
| [124] | Niijima, K.: Pointwise error estimates for a streamline diffusion finite element scheme. Numer. Math. 56(7), 707-719 (1990) · Zbl 0691.65077 |
| [125] | 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) · Zbl 1329.65278 |
| [126] | 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) · Zbl 0844.65075 |
| [127] | 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) · Zbl 1155.65087 |
| [128] | Saad, Y.: A flexible inner – outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461-469 (1993) · Zbl 0780.65022 |
| [129] | Sangalli, G.: Robust a-posteriori estimator for advection – diffusion – reaction problems. Math. Comput. 77(261), 41-70 (2008). (electronic) · Zbl 1130.65083 |
| [130] | Schötzau, D., Schwab, C., Stenberg, R.: Mixed \[hp\] hp-FEM on anisotropic meshes. II. Hanging nodes and tensor products of boundary layer meshes. Numer. Math. 83(4), 667-697 (1999) · Zbl 0958.76049 |
| [131] | 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) · Zbl 1388.76151 |
| [132] | Schwegler, K., Bause, M.: Goal-oriented a posteriori error control for nonstationary convection-dominated transport problems. Technical Report arXiv:1601.06544 (2016) |
| [133] | 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) · Zbl 0582.76028 |
| [134] | 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) · Zbl 1253.65026 |
| [135] | 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) · Zbl 1306.76026 |
| [136] | Tobiska, L., Verfürth, R.: Robust a posteriori error estimates for stabilized finite element methods. IMA J. Numer. Anal. 35(4), 1652-1671 (2015) · Zbl 1334.65186 |
| [137] | Vanka, S.P.: Block-implicit multigrid solution of Navier-Stokes equations in primitive variables. J. Comput. Phys. 65(1), 138-158 (1986) · Zbl 0606.76035 |
| [138] | Verfürth, R.: A posteriori error estimators for convection – diffusion equations. Numer. Math. 80(4), 641-663 (1998) · Zbl 0913.65095 |
| [139] | Verfürth, R.: Robust a posteriori error estimates for stationary convection – diffusion equations. SIAM J. Numer. Anal. 43(4), 1766-1782 (2005). (electronic) · Zbl 1099.65100 |
| [140] | 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) · Zbl 1375.65158 |
| [141] | Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithms for fluids. J. Comput. Phys. 31(3), 335-362 (1979) · Zbl 0416.76002 |
| [142] | Zhang, S.: A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74(250), 543-554 (2005) · Zbl 1085.76042 |
| [143] | Zhou, G.H., Rannacher, R.: Pointwise superconvergence of the streamline diffusion finite-element method. Numer. Methods Partial Differ. Equ 12(1), 123-145 (1996) · Zbl 0841.65092 |
| [144] | Zhou, G.: How accurate is the streamline diffusion finite element method? Math. Comput. 66(217), 31-44 (1997) · Zbl 0854.65094 |
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