×

Analysis of fully discrete, quasi non-conforming approximations of evolution equations and applications. (English) Zbl 1493.47119

Summary: In this paper, we consider fully discrete approximations of abstract evolution equations, by means of a quasi non-conforming spatial approximation and finite differences in time (Rothe-Galerkin method). The main result is the convergence of the discrete solutions to a weak solution of the continuous problem. Hence, the result can be interpreted either as a justification of the numerical method or as an alternative way of constructing weak solutions. We set the problem in the very general and abstract setting of pseudo-monotone operators, which allows for a unified treatment of several evolution problems. The examples – which fit into our setting and which motivated our research – are problems describing the motion of incompressible fluids, since the quasi non-conforming approximation allows to handle problems with prescribed divergence. Our abstract results for pseudo-monotone operators allow to show convergence just by verifying a few natural assumptions on the operator time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be easily performed. The results of some numerical experiments are reported in the final section.

MSC:

47J35 Nonlinear evolution equations
47H05 Monotone operators and generalizations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K90 Abstract parabolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence

Software:

FEniCS; Matplotlib; SyFi
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alt, H. and Luckhaus, S., Quasilinear elliptic-parabolic differential equations, Math. Z.183 (1983) 311-341. · Zbl 0497.35049
[2] Amrouche, C., Berselli, L., Lewandowski, R. and Nguyen, D., Turbulent flows as generalized Kelvin-Voigt materials: Modeling and analysis, Nonlinear Anal.196 (2020) 111790, 24 pp. · Zbl 1434.76027
[3] Barrett, J. W. and Liu, W. B., Finite element approximation of the parabolic \(p\)-Laplacian, SIAM J. Numer. Anal.31 (1994) 413-428. · Zbl 0805.65097
[4] Bartels, S., Numerical Approximation of Partial Differential Equations, , Vol. 64 (Springer, 2016). · Zbl 1353.65089
[5] Bartels, S., Diening, L. and Nochetto, R., Unconditional stability of semi-implicit discretizations of singular flows, SIAM J. Numer. Anal.56 (2018) 1896-1914. · Zbl 1394.65098
[6] Bartels, S., Nochetto, R. and Salgado, A., Discrete total variation flows without regularization, SIAM J. Numer. Anal.52 (2014) 363-385. · Zbl 1292.65106
[7] Bartels, S. and Růžička, M., Convergence of fully discrete implicit and semi-implicit approximations of singular parabolic equations, SIAM J. Numer. Anal.58 (2020) 811-833. · Zbl 1447.65068
[8] Bäumle, E. and Růžička, M., Existence of weak solutions for unsteady motions of micro-polar electrorheological fluids, SIAM J. Math. Anal.49 (2017) 115-141. · Zbl 1457.35033
[9] Belenki, L., Berselli, L. C., Diening, L. and Růžička, M., On the finite element approximation of \(p\)-Stokes systems, SIAM J. Numer. Anal.50 (2012) 373-397. · Zbl 1426.76221
[10] Berselli, L. C., Diening, L. and Růžička, M., Optimal error estimates for semi-implicit space-time discretization for the equations describing incompressible generalized Newtonian fluids, IMA J. Num. Anal.25 (2015) 680-697. · Zbl 1326.76056
[11] Berselli, L. C. and Růžička, M., Space-time discretization for nonlinear parabolic systems with \(p\)-structure, IMA J. Numer. Anal. (2020), https://doi.org/10.1093/imanum/draa079. · Zbl 1417.76004
[12] Breit, D. and Mensah, P., Space-time approximation of parabolic systems with variable growth, IMA J. Numer. Anal.40 (2019) 2505-2552. · Zbl 1466.65122
[13] Brenner, S. and Scott, L., The Mathematical Theory of Finite Element Methods, 3rd edn. , Vol. 15 (Springer, 2008). · Zbl 1135.65042
[14] Brezzi, F. and Fortin, M., Mixed and Hybrid Finite Element Methods, , Vol. 15 (Springer-Verlag, 1991). · Zbl 0788.73002
[15] Carelli, E., Haehnle, J. and Prohl, A., Convergence analysis for incompressible generalized Newtonian fluid flows with nonstandard anisotropic growth conditions, SIAM J. Numer. Anal.48 (2010) 164-190. · Zbl 1428.35347
[16] Rebollo, T. Chacón and Lewandowski, R., Mathematical and Numerical Foundations of Turbulence Models and Applications, (Birkhäuser/Springer, 2014). · Zbl 1328.76002
[17] Clément, P., Approximation by finite element functions using local regularization, RAIRO Anal. Numér.9 (1975) 77-84. · Zbl 0368.65008
[18] Di Pietro, D. and Ern, A., Mathematical Aspects of Discontinuous Galerkin Methods, , Vol. 69 (Springer, 2012). · Zbl 1231.65209
[19] Diening, L., Ebmeyer, C. and Růžička, M., Optimal convergence for the implicit space-time discretization of parabolic systems with \(p\)-structure, SIAM J. Numer. Anal.45 (2007) 457-472. · Zbl 1140.65060
[20] Diening, L., Růžička, M. and Schumacher, K., A decomposition technique for John domains, Ann. Acad. Sci. Fenn. Math.35 (2010) 87-114. · Zbl 1194.26022
[21] L. Diening, J. Storn and T. Tscherpel, Fortin operator for the Taylor-Hood element, preprint (2021), arXiv:2104.13953 [math.NA]. · Zbl 1477.65207
[22] Durán, R., Otárola, E. and Salgado, A., Stability of the Stokes projection on weighted spaces and applications, Math. Comp.89 (2020) 1581-1603. · Zbl 1437.35572
[23] Dyda, B., Ihnatsyeva, L., Lehrbäck, J., Tuominen, H. and Vähäkangas, A., Muckenhoupt \(A_p\)-properties of distance functions and applications to Hardy-Sobolev-type inequalities, Potential Anal.50 (2019) 83-105. · Zbl 1405.42031
[24] Eckart, W. and Růžička, M., Modeling micropolar electrorheological fluids, Int. J. Appl. Mech. Eng.11 (2006) 813-844. · Zbl 1196.76015
[25] Eckstein, S. and Růžička, M., On the full space-time discretization of the generalized Stokes equations: The Dirichlet case, SIAM J. Numer. Anal.56 (2018) 2234-2261. · Zbl 1433.65184
[26] Feng, X., von Oehsen, M. and Prohl, A., Rate of convergence of regularization procedures and finite element approximations for the total variation flow, Numer. Math.100 (2005) 441-456. · Zbl 1075.65114
[27] A. Fröhlich, Stokes- und Navier-Stokes-Gleichungen in gewichteten Funktionenräumen, phD Thesis, FB Mathematik Universität Darmstadt, Shaker Verlag, Aachen (2001). · Zbl 0981.35050
[28] Gajewski, H., Gröger, K. and Zacharias, K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie-Verlag, 1974). · Zbl 0289.47029
[29] Girault, V. and Lions, J.-L., Two-grid finite-element schemes for the transient Navier-Stokes problem, M2AN Math. Model. Numer. Anal.35 (2001) 945-980. · Zbl 1032.76032
[30] Girault, V., Nochetto, R. H. and Scott, L. R., Max-norm estimates for Stokes and Navier-Stokes approximations in convex polyhedra, Numer. Math.131 (2015) 771-822. · Zbl 1401.76087
[31] Girault, V. and Scott, L. R., A quasi-local interpolation operator preserving the discrete divergence, Calcolo40 (2003) 1-19. · Zbl 1072.65014
[32] Heinonen, J., Kilpeläinen, T. and Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations, (The Clarendon Press, Oxford University Press, New York, 1993). · Zbl 0780.31001
[33] Hunter, J. D., Matplotlib: A 2d graphics environment, Comput. Sci. Eng.9 (2007) 90-95.
[34] A. Kaltenbach, Note on the existence theory for non-induced evolution problems, accepted Math. Nach. (2019).
[35] Kaltenbach, A. and Růžička, M., Note on the existence theory for pseudo-monotone evolution problems, J. Evol. Equ.21 (2021) 247-276. · Zbl 07340071
[36] Kufner, A. and Opic, B., How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin.25 (1984) 537-554. · Zbl 0557.46025
[37] A. Logg, G. Wells and K.-A. Mardal, Automated solution of differential equations by the finite element method, The FEniCS Book, Vol. 84 (2011).
[38] Łukaszewicz, G., Long time behavior of 2D micropolar fluid flows, Math. Comput. Model.34 (2001) 487-509. · Zbl 1020.76003
[39] Nochetto, R., Otárola, E. and Salgado, A., Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications, Numer. Math.132 (2016) 85-130. · Zbl 1334.65030
[40] Nochetto, R., Salgado, A. and Tomas, I., The micropolar Navier-Stokes equations: A priori error analysis, Math. Models Methods Appl. Sci.24 (2014) 1237-1264. · Zbl 1288.76019
[41] Nochetto, R. H., Savaré, G. and Verdi, C., A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations, Comm. Pure Appl. Math.53 (2000) 525-589. · Zbl 1021.65047
[42] Prohl, A., Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamics system, M2AN Math. Model. Numer. Anal.42 (2008) 1065-1087. · Zbl 1149.76029
[43] Ravindran, S., Analysis of a decoupled time-stepping scheme for evolutionary micropolar fluid flows, Adv. Numer. Anal. (2016) Article ID:7010645, 13 pp. · Zbl 1422.76137
[44] Roubíček, T., Nonlinear Partial Differential Equations with Applications, , Vol. 153 (Birkhäuser, 2005). · Zbl 1087.35002
[45] Rulla, J., Error analysis for implicit approximations to solutions to Cauchy problems, SIAM J. Numer. Anal.33 (1996) 68-87. · Zbl 0855.65102
[46] Růžička, M., Electrorheological Fluids: Modeling and Mathematical Theory, , Vol. 1748 (Springer, 2000). · Zbl 0962.76001
[47] Sagaut, P., Large Eddy Simulation for Incompressible Flows: An Introduction, , 2nd edn. (Springer-Verlag, 2002), translated from the 1998 French original, with a foreword by Marcel Lesieur, with a foreword by Massimo Germano.
[48] L. R. Scott, A local Fortin operator for low-order Taylor-Hood elements, preprint TR-2021-07, University of Chicago (2021).
[49] Scott, L. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp.54 (1990) 483-493. · Zbl 0696.65007
[50] Showalter, R., Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, (Amer. Math. Soc., 1997). · Zbl 0870.35004
[51] Süli, E. and Tscherpel, T., Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids, IMA J. Numer. Anal.40 (2019) 801-849. · Zbl 1464.65131
[52] Temam, R., Une méthode d’approximation de la solution des équations de Navier-Stokes, Bull. Soc. Math. France96 (1968) 115-152. · Zbl 0181.18903
[53] Temam, R., Navier-Stokes Equations. Theory and Numerical Analysis, , Vol. 2 (North-Holland Publishing Co., 1977). · Zbl 0383.35057
[54] T. Tscherpel, Finite element approximation for the unsteady flow of implicitly constituted incompressible fluids, Ph.D. Thesis, University of Oxford (2018). · Zbl 1464.65131
[55] Zeidler, E., Nonlinear Functional Analysis and its Applications: II/A Linear Monotone Operators (Springer-Verlag, 1990). · Zbl 0684.47028
[56] Zeidler, E., Nonlinear Functional Analysis and its Applications: II/B Nonlinear Monotone Operators (Springer-Verlag, 1990). · Zbl 0684.47029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.