A class of space-time discretizations for the stochastic \(p\)-Stokes system. (English) Zbl 07923226
Summary: The main objective of the present paper is to construct a new class of space-time discretizations for the stochastic \(p\)-Stokes system and analyze its stability and convergence properties.
We derive regularity results for the approximation that are similar to the natural regularity of solutions. One of the key arguments relies on discrete extrapolation that allows us to relate lower moments of discrete maximal processes.
We show that, if the generic spatial discretization is constraint conforming, then the velocity approximation satisfies a best-approximation property in the natural distance.
Moreover, we present an example such that the resulting velocity approximation converges with rate \(1 / 2\) in time and 1 in space towards the (unknown) target velocity with respect to the natural distance. The theory is corroborated by numerical experiments.
We derive regularity results for the approximation that are similar to the natural regularity of solutions. One of the key arguments relies on discrete extrapolation that allows us to relate lower moments of discrete maximal processes.
We show that, if the generic spatial discretization is constraint conforming, then the velocity approximation satisfies a best-approximation property in the natural distance.
Moreover, we present an example such that the resulting velocity approximation converges with rate \(1 / 2\) in time and 1 in space towards the (unknown) target velocity with respect to the natural distance. The theory is corroborated by numerical experiments.
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35K65 | Degenerate parabolic equations |
| 35K67 | Singular parabolic equations |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 34K28 | Numerical approximation of solutions of functional-differential equations (MSC2010) |
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
Keywords:
SPDEs; stochastic \(p\)-Stokes system; power-law fluids; conforming finite element methods; convergence rates; error analysisReferences:
| [1] | Ahlkrona, J.; Braack, M., Equal-order stabilized finite element approximation of the \(p\)-Stokes equations on anisotropic cartesian meshes, Comput. Methods Appl. Math., 20, 1, 1-25, 2020 · Zbl 1437.65175 |
| [2] | Balay, S.; Abhyankar, S.; Adams, M. F.; Benson, S.; Brown, J.; Brune, P.; Buschelman, K.; Constantinescu, E. M.; Dalcin, L.; Dener, A.; Eijkhout, V.; Faibussowitsch, J.; Gropp, W. D.; Hapla, V.; Isaac, T.; Jolivet, P.; Karpeev, D.; Kaushik, D.; Knepley, M. G.; Kong, F.; Kruger, S.; May, D. A.; McInnes, L. C.; Mills, R. T.; Mitchell, L.; Munson, T.; Roman, J. E.; Rupp, K.; Sanan, P.; Sarich, J.; Smith, B. F.; Zampini, S.; Zhang, H.; Zhang, H.; Zhang, J., PETSc web page, 2024, https://petsc.org/ |
| [3] | Baňas, Ľ.; Röckner, M.; Wilke, A., Convergent numerical approximation of the stochastic total variation flow, Stoch. Partial Differ. Equ. Anal. Comput., 9, 2, 437-471, 2021 · Zbl 1468.60075 |
| [4] | Baranger, J.; Najib, K., Analyse numérique des écoulements quasi-newtoniens dont la viscosité obéit à la loi puissance ou la loi de carreau, Numer. Math., 58, 1, 35-49, 1990 · Zbl 0702.76007 |
| [5] | Barrett, J. W.; Liu, W. B., Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow, Numer. Math., 68, 4, 437-456, 1994 · Zbl 0811.76036 |
| [6] | Bauzet, C.; Nabet, F.; Schmitz, K.; Zimmermann, A., Convergence of a finite-volume scheme for a heat equation with a multiplicative Lipschitz noise, ESAIM Math. Model. Numer. Anal., 57, 2, 745-783, 2023 · Zbl 1515.60231 |
| [7] | Becker, S.; Gess, B.; Jentzen, A.; Kloeden, P. E., Lower and upper bounds for strong approximation errors for numerical approximations of stochastic heat equations, BIT, 60, 4, 1057-1073, 2020 · Zbl 1472.60111 |
| [8] | Belenki, L.; Berselli, L. C.; Diening, L.; Ružička, M., On the finite element approximation of \(p\)-Stokes systems, SIAM J. Numer. Anal., 50, 2, 373-397, 2012 · Zbl 1426.76221 |
| [9] | Berselli, L. C.; Diening, L.; Růžička, M., Optimal error estimates for a semi-implicit Euler scheme for incompressible fluids with shear dependent viscosities, SIAM J. Numer. Anal., 47, 3, 2177-2202, 2009 · Zbl 1406.76006 |
| [10] | Berselli, L. C.; Kaltenbach, A.; Růžička, M., Analysis of fully discrete, quasi non-conforming approximations of evolution equations and applications, Math. Models Methods Appl. Sci., 31, 11, 2297-2343, 2021 · Zbl 1493.47119 |
| [11] | Botti, M.; Castanon Quiroz, D.; Di Pietro, D. A.; Harnist, A., A hybrid high-order method for creeping flows of non-Newtonian fluids, ESAIM Math. Model. Numer. Anal., 55, 5, 2045-2073, 2021 · Zbl 1489.65154 |
| [12] | Breit, D., Existence theory for stochastic power law fluids, J. Math. Fluid Mech., 17, 2, 295-326, 2015 · Zbl 1327.35461 |
| [13] | Breit, D.; Diening, L.; Storn, J.; Wichmann, J., The parabolic \(p\)-Laplacian with fractional differentiability, IMA J. Numer. Anal., 41, 3, 2110-2138, 2021 · Zbl 07528300 |
| [14] | Breit, D.; Feireisl, E.; Hofmanová, M., Compressible fluids driven by stochastic forcing: the relative energy inequality and applications, Comm. Math. Phys., 350, 2, 443-473, 2017 · Zbl 1362.35228 |
| [15] | Breit, D.; Gmeineder, F., Electro-rheological fluids under random influences: martingale and strong solutions, Stoch. Partial Differ. Equ. Anal. Comput., 7, 4, 699-745, 2019 · Zbl 1431.60054 |
| [16] | Breit, D.; Hofmanová, M.; Loisel, S., Space-time approximation of stochastic \(p\)-Laplace-type systems, SIAM J. Numer. Anal., 59, 4, 2218-2236, 2021 · Zbl 1496.65156 |
| [17] | Brzeźniak, Z.; Carelli, E.; Prohl, A., Finite-element-based discretizations of the incompressible Navier-Stokes equations with multiplicative random forcing, IMA J. Numer. Anal., 33, 3, 771-824, 2013 · Zbl 1426.76227 |
| [18] | Carelli, E.; Haehnle, J.; Prohl, A., Convergence analysis for incompressible generalized Newtonian fluid flows with nonstandard anisotropic growth conditions, SIAM J. Numer. Anal., 48, 1, 164-190, 2010 · Zbl 1428.35347 |
| [19] | Carstensen, C.; Gallistl, D.; Schedensack, M., Quasi-optimal adaptive pseudostress approximation of the Stokes equations, SIAM J. Numer. Anal., 51, 3, 1715-1734, 2013 · Zbl 1383.76333 |
| [20] | Carstensen, C.; Peterseim, D.; Rabus, H., Optimal adaptive nonconforming FEM for the Stokes problem, Numer. Math., 123, 2, 291-308, 2013 · Zbl 1316.76046 |
| [21] | Castanon Quiroz, D.; Di Pietro, D. A.; Harnist, A., A hybrid high-order method for incompressible flows of non-Newtonian fluids with power-like convective behaviour, IMA J. Numer. Anal., 43, 1, 144-186, 2023 · Zbl 07658046 |
| [22] | de Diego, G. G.; Farrell, P. E.; Hewitt, I. J., On the finite element approximation of a semicoercive Stokes variational inequality arising in glaciology, SIAM J. Numer. Anal., 61, 1, 1-25, 2023 · Zbl 1521.65124 |
| [23] | Di Pietro, D. A.; Droniou, J.; Harnist, A., Improved error estimates for hybrid high-order discretizations of Leray-Lions problems, Calcolo, 58, 2, 24, 2021, Paper No. 19 · Zbl 1471.65176 |
| [24] | Diening, L.; Fornasier, M.; Tomasi, R.; Wank, M., A relaxed Kaanov iteration for the p-poisson problem, Numer. Math., 2020 |
| [25] | Diening, L.; Harjulehto, P.; Hästö, P.; Růžička, M., (Lebesgue and Sobolev Spaces with Variable Exponents. Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, 2011, Springer: Springer Heidelberg), x+509 · Zbl 1222.46002 |
| [26] | Diening, L.; Hofmanová, M.; Wichmann, J., An averaged space-time discretization of the stochastic p-Laplace system, Numer. Math., 2022 |
| [27] | Diening, L.; Kreuzer, C.; Süli, E., Finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology, SIAM J. Numer. Anal., 51, 2, 984-1015, 2013 · Zbl 1268.76030 |
| [28] | Diening, L.; Storn, J.; Tscherpel, T., On the Sobolev and \(L^p\)-stability of the \(L^2\)-projection, SIAM J. Numer. Anal., 59, 5, 2571-2607, 2021 · Zbl 1477.65207 |
| [29] | Droniou, J.; Eymard, R.; Gallouët, T.; Herbin, R., A unified analysis of elliptic problems with various boundary conditions and their approximation, Czechoslovak Math. J., 70, 145, 339-368, 2020 · Zbl 1513.65498 |
| [30] | Droniou, J.; Goldys, B.; Le, K.-N., Design and convergence analysis of numerical methods for stochastic evolution equations with Leray-Lions operator, IMA J. Numer. Anal., 42, 2, 1143-1179, 2022 · Zbl 1514.65010 |
| [31] | Eckstein, S.; Růžička, M., On the full space-time discretization of the generalized Stokes equations: the Dirichlet case, SIAM J. Numer. Anal., 56, 4, 2234-2261, 2018 · Zbl 1433.65184 |
| [32] | Eisenmann, M.; Kovács, M.; Kruse, R.; Larsson, S., On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients, Found. Comput. Math., 19, 6, 1387-1430, 2019 · Zbl 07135262 |
| [33] | Emmrich, E., Convergence of a time discretization for a class of non-Newtonian fluid flow, Commun. Math. Sci., 6, 4, 827-843, 2008, URL http://projecteuclid.org/euclid.cms/1229619672 · Zbl 1160.76034 |
| [34] | Emmrich, E.; Šiška, D., Nonlinear stochastic evolution equations of second order with damping, Stoch. Partial Differ. Equ. Anal. Comput., 5, 1, 81-112, 2017 · Zbl 1362.60061 |
| [35] | Feng, X.; Prohl, A.; Vo, L., Optimally convergent mixed finite element methods for the stochastic Stokes equations, IMA J. Numer. Anal., 41, 3, 2280-2310, 2021 · Zbl 1492.65022 |
| [36] | Feng, X.; Qiu, H., Analysis of fully discrete mixed finite element methods for time-dependent stochastic Stokes equations with multiplicative noise, J. Sci. Comput., 88, 2, 25, 2021, Paper No. 31 · Zbl 1491.65092 |
| [37] | Flandoli, F., An introduction to 3D stochastic fluid dynamics, (SPDE in Hydrodynamic: Recent Progress and Prospects. SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Math., vol. 1942, 2008, Springer: Springer Berlin), 51-150 · Zbl 1426.76001 |
| [38] | Guzmán, J.; Neilan, M., Conforming and divergence-free Stokes elements in three dimensions, IMA J. Numer. Anal., 34, 4, 1489-1508, 2014 · Zbl 1305.76056 |
| [39] | Guzmán, J.; Neilan, M., Conforming and divergence-free Stokes elements on general triangular meshes, Math. Comp., 83, 285, 15-36, 2014 · Zbl 1322.76041 |
| [40] | Guzmán, J.; Scott, L. R., Cubic Lagrange elements satisfying exact incompressibility, SMAI J. Comput. Math., 4, 345-374, 2018 · Zbl 1416.76109 |
| [41] | Guzmán, J.; Scott, L. R., The Scott-Vogelius finite elements revisited, Math. Comp., 88, 316, 515-529, 2019 · Zbl 1405.65150 |
| [42] | Gyöngy, I.; Millet, A., On discretization schemes for stochastic evolution equations, Potential Anal., 23, 2, 99-134, 2005 · Zbl 1067.60049 |
| [43] | Gyöngy, I.; Millet, A., Rate of convergence of space time approximations for stochastic evolution equations, Potential Anal., 30, 1, 29-64, 2009 · Zbl 1168.60025 |
| [44] | Ham, D. A.; Kelly, P. H.J.; Mitchell, L.; Cotter, C. J.; Kirby, R. C.; Sagiyama, K.; Bouziani, N.; Vorderwuelbecke, S.; Gregory, T. J.; Betteridge, J.; Shapero, D. R.; Nixon-Hill, R. W.; Ward, C. J.; Farrell, P. E.; Brubeck, P. D.; Marsden, I.; Gibson, T. H.; Homolya, M.; Sun, T.; McRae, A. T.T.; Luporini, F.; Gregory, A.; Lange, M.; Funke, S. W.; Rathgeber, F.; Bercea, G.-T.; Markall, G. R., Firedrake User Manual, 2023, Imperial College London and University of Oxford and Baylor University and University of Washington |
| [45] | Hirn, A., Approximation of the \(p\)-Stokes equations with equal-order finite elements, J. Math. Fluid Mech., 15, 1, 65-88, 2013 · Zbl 1430.76365 |
| [46] | Hytönen, T. P.; Veraar, M. C., On besov regularity of Brownian motions in infinite dimensions, Probab. Math. Statist., 28, 1, 143-162, 2008 · Zbl 1136.60358 |
| [47] | Jentzen, A.; Kloeden, P. E., The numerical approximation of stochastic partial differential equations, Milan J. Math., 77, 205-244, 2009 · Zbl 1205.60130 |
| [48] | Jentzen, A.; Kurniawan, R., Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients, Found. Comput. Math., 21, 2, 445-536, 2021 · Zbl 1482.65143 |
| [49] | Kaltenbach, A.; Růžička, M., A local discontinuous Galerkin approximation for the \(p\)-Navier-Stokes system, part II: Convergence rates for the velocity, SIAM J. Numer. Anal., 61, 4, 1641-1663, 2023 · Zbl 1518.76036 |
| [50] | Kaltenbach, A.; Růžička, M., A local discontinuous Galerkin approximation for the \(p\)-Navier-Stokes system, part III: Convergence rates for the pressure, SIAM J. Numer. Anal., 61, 4, 1763-1782, 2023 · Zbl 1518.76037 |
| [51] | Kamrani, M.; Hosseini, S. M.; Hausenblas, E., Implicit Euler method for numerical solution of nonlinear stochastic partial differential equations with multiplicative trace class noise, Math. Methods Appl. Sci., 41, 13, 4986-5002, 2018 · Zbl 1394.65085 |
| [52] | Klioba, K.; Veraar, M., Pathwise uniform convergence of time discretisation schemes for SPDEs, 2023, arXiv e-prints, arXiv:2303.00411. arXiv:2303.00411 [math.NA] |
| [53] | Málek, J.; Rajagopal, K. R., Mathematical issues concerning the Navier-Stokes equations and some of its generalizations, (Evolutionary Equations. Vol. II. Evolutionary Equations. Vol. II, Handb. Differ. Equ., 2005, Elsevier/North-Holland, Amsterdam), 371-459 · Zbl 1095.35027 |
| [54] | Mikulevicius, R.; Rozovskii, B. L., Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35, 5, 1250-1310, 2004 · Zbl 1062.60061 |
| [55] | Ondreját, M.; Prohl, A.; Walkington, N. J., Numerical approximation of nonlinear SPDE’s, (Stochastics and Partial Differential Equations: Analysis and Computations, 2022, Springer Science and Business Media LLC) |
| [56] | Ondreját, M.; Veraar, M., On temporal regularity of stochastic convolutions in 2-smooth Banach spaces, Ann. Inst. Henri Poincaré Probab. Stat., 56, 3, 1792-1808, 2020 · Zbl 1483.60058 |
| [57] | Prohl, A.; Ružička, M., On fully implicit space-time discretization for motions of incompressible fluids with shear-dependent viscosities: the case \(p \leq 2\), SIAM J. Numer. Anal., 39, 1, 214-249, 2001 · Zbl 1008.76047 |
| [58] | Revuz, D.; Yor, M., (Continuous Martingales and Brownian Motion. Continuous Martingales and Brownian Motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 1999, Springer-Verlag: Springer-Verlag Berlin), xiv+602 · Zbl 0917.60006 |
| [59] | Scott, L. R.; Vogelius, M., Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO Modél. Math. Anal. Numér., 19, 1, 111-143, 1985 · Zbl 0608.65013 |
| [60] | Simon, J., Sobolev, Besov and Nikolskii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval, Ann. Mat. Pura Appl. (4), 157, 117-148, 1990 · Zbl 0727.46018 |
| [61] | Süli, E.; Tscherpel, T., Fully discrete finite element approximation of unsteady flows of implicitly constituted incompressible fluids, IMA J. Numer. Anal., 40, 2, 801-849, 2020 · Zbl 1464.65131 |
| [62] | Taylor, C.; Hood, P., A numerical solution of the Navier-Stokes equations using the finite element technique, Comput. & Fluids, 1, 1, 73-100, 1973 · Zbl 0328.76020 |
| [63] | Tscherpel, T., Finite Element Approximation for the Unsteady Flow of Implicitly Constituted Incompressible Fluids, 2018, University of Oxford, (Ph.D. thesis) |
| [64] | Wichmann, J., Temporal regularity of symmetric stochastic p-Stokes systems, J. Math. Fluid Mech., 26, 2, 2024 · Zbl 1537.35133 |
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.
