An adaptive multiscale hybrid-mixed method for the Oseen equations

Springer Science and Business Media LLC - Tập 47 - Trang 1-36 - 2021
Rodolfo Araya1, Cristian Cárcamo1,2, Abner H. Poza3, Frédéric Valentin4
1Departamento de Ingeniería Matemática & CI2MA, Universidad de Concepción, Concepción, Chile
2Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Groningen, The Netherlands
3Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Concepción, Chile
4Department of Mathematical and Computational Methods, LNCC - National Laboratory for Scientific Computing, Petrópolis, Brazil

Tóm tắt

A novel residual a posteriori error estimator for the Oseen equations achieves efficiency and reliability by including multilevel contributions in its construction. Originates from the Multiscale Hybrid Mixed (MHM) method, the estimator combines residuals from the skeleton of the first-level partition of the domain, along with the contributions from element-wise approximations. The second-level estimator is local and infers the accuracy of multiscale basis computations as part of the MHM framework. Also, the face-degrees of freedom of the MHM method shape the estimator and induce a new face-adaptive procedure on the mesh’s skeleton only. As a result, the approach avoids re-meshing the first-level partition, which makes the adaptive process affordable and straightforward on complex geometries. Several numerical tests assess theoretical results.

Tài liệu tham khảo

Abdulle, A., Nonnenmacher, A.: A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method. Numer. Methods Partial Differential Equations 29(5), 1629–1656 (2013). https://doi.org/10.1002/num.21769 Agmon, S.: Lectures on Elliptic Boundary Problems. AMS Chelsea Publishing, Providence (2010). https://doi.org/10.1090/chel/369 Ahmed, N., Chacón, T., John, V., Rubino, S.: A review of variational multiscale methods for the simulation of turbulent incompressible flows. Arch. Comput. Methods Eng. 24(1), 115–164 (2017). https://doi.org/10.1007/s11831-015-9161-0 Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York (2000). https://doi.org/10.1002/9781118032824 Araya, R., Barrenechea, G.R., Poza, A.H., Valentin, F.: Convergence analysis of a residual local projection finite element method for the Navier-Stokes equations. SIAM J. Numer. Anal. 50(2), 669–699 (2012). https://doi.org/10.1137/110829283 Araya, R., Barrenechea, G.R., Valentin, F.: Stabilized finite element methods based on multiscale enrichment for the Stokes problem. SIAM J. Numer. Anal. 44(1), 322–348 (2006) Araya, R., Bertoglio, C., Cárcamo, C., Poza, A.H., Valentin, F.: Multiscale hybrid-mixed method for the Oseen equation. In preparation (2021) Araya, R., Harder, C., Paredes, D., Valentin, F.: Multiscale hybrid-mixed method. SIAM J. Numer. Anal. 51(6), 3505–3531 (2013). https://doi.org/10.1137/120888223 Araya, R., Harder, C., Poza, A.H., Valentin, F.: Multiscale hybrid-mixed method for the Stokes and Brinkman equations - The method. Comput. Methods Appl. Mech. Engrg. 324(1), 172–197 (2017). https://doi.org/10.1016/j.cma.2017.05.027 Araya, R., Poza, A.H., Valentin, F.: An adaptive residual local projection finite element method for the Navier–Stokes equations. Adv. Comput. Math. 40(5), 1093–1119 (2014). https://doi.org/10.1007/s10444-014-9343-6 Araya, R., Rebolledo, R., Valentin, F.: On a multiscale a posteriori error estimator for the Stokes and Brinkman equations. IMA J. Numer Anal. https://doi.org/10.1093/imanum/drz053 (2019) Arbogast, T., Pencheva, G., Wheeler, M.F., Yotov, I.: A multiscale mortar mixed finite element method. Multiscale Model Simul. 6(1), 319–346 (2007). https://doi.org/10.1137/060662587 Babuska, I., Osborn, E.: Generalized finite element methods: their performance and their relation to mixed methods. SIAM J. Num. Anal. 20(3), 510–536 (1983) Baiges, J., Codina, R.: Variational multiscale error estimators for solid mechanics adaptive simulations: an orthogonal subgrid scale approach. Comput. Methods Appl. Mech. Engrg. 325, 37–55 (2017). https://doi.org/10.1016/j.cma.2017.07.008 Barrenechea, G., Fernández, M., Vidal, C.: A stabilized finite element method for the Oseen equation with dominating reaction. Research Report RR-5213, INRIA. https://hal.inria.fr/inria-00070780 (2004) Barrenechea, G.R., Franca, L.P., Valentin, F.: A Petrov-Galerkin enriched method: a mass conservative finite element method for the Darcy equation. Comput. Methods Appl. Mech. Engrg. 196(21-24), 2449–2464 (2007). https://doi.org/10.1016/j.cma.2007.01.004 Barrenechea, G.R., Valentin, F.: A residual local projection method for the Oseen equation. Comput. Methods Appl. Mech. Engrg. 199(29-32), 1906–1921 (2010). https://doi.org/10.1016/j.cma.2010.01.014 Bi, C., Wang, C., Lin, Y.: Two-grid finite element method and its a posteriori error estimates for a nonmonotone quasilinear elliptic problem under minimal regularity of data. Comput. Math. Appl. 76(1), 98–112 (2018). https://doi.org/10.1016/j.camwa.2018.04.006 Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Texts in Applied Mathematics, 3rd edn., vol. 15. Springer, New York (2008). https://doi.org/10.1007/978-0-387-75934-0 Cesmelioglu, A., Cockburn, B., Nguyen, N.C., Peraire, J.: Analysis of HDG methods for Oseen equations. J. Sci. Comput. 55(2), 392–431 (2013). https://doi.org/10.1007/s10915-012-9639-y Chamoin, L., Legoll, F.: A posteriori error estimation and adaptive strategy for the control of msFEM computations. Comput. Methods Appl. Mech. Engrg. 336, 1–38 (2018). https://doi.org/10.1016/j.cma.2018.02.016 Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002) Clément, P.: Approximation by finite element functions using local regularization. R.A.I.R.O Anal. Numer. 9(R-2), 77–84 (1975) Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319–1365 (2009). https://doi.org/10.1137/070706616 Davidson, P.A.: Turbulence. An Introduction for Scientists and Engineers, 2nd edn. Oxford University Press, Oxford (2015). https://doi.org/10.1093/acprof:oso/9780198722588.001.0001 Durán, R.: An elementary proof of the continuity from \({L}^{2}_{0}({{\Omega }})\) to \({H}^{1}_{0}({{\Omega }})\) of Bogovskii’s right inverse of the divergence. Rev. Un. Mat. Argentina 53(2), 59–78 (2012) E, W., Engquist, B., Li, X., Ren, W., Vanden-Eijnden, E.: Heterogeneous multiscale methods: a review. Commun. Comput. Phys. 2(3), 367–450 (2007) Efendiev, Y., Galvis, J., Hou, T.Y.: Generalized multiscale finite element methods (GMsFEM). J. Comput. Phys. 251, 116–135 (2013). https://doi.org/10.1016/j.jcp.2013.04.045 Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004). https://doi.org/10.1007/978-1-4757-4355-5 Ervin, V., Layton, W., Maubach, J.: A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations. Numer. Methods Partial Differential Equations 12(3), 333–346 (1996). https://doi.org/10.1002/(SICI)1098-2426(199605)12:3<333::AID-NUM4>3.0.CO;2-P Farhat, C., Harari, I., Franca, L.P.: The discontinuous enrichment method. Comput. Methods Appl. Mech. Engrg. 190(48), 6455–6479 (2001). https://doi.org/10.1016/S0045-7825(01)00232-8 Ferziger, J.H., Perić, M., Street, R.L.: Computational Methods for Fluid Dynamics, 4th edn. Springer, Cham (2020). https://doi.org/10.1007/978-3-319-99693-6 Foias, C., Manley, O., Rosa, R., Temam, R.: Navier-Stokes Equations and Turbulence. Encyclopedia of Mathematics and its Applications, vol. 83. Cambridge University Press, Cambridge (2001). https://doi.org/10.1017/CBO9780511546754 Franca, L.P., Harder, C., Valentin, F.: On a residual local projection method for the Darcy equation. C. R. Math. Acad. Sci. Paris 347(17-18), 1105–1110 (2009). https://doi.org/10.1016/j.crma.2009.06.016 Franca, L.P., Madureira, A.L.: Element diameter free stability parameters for stabilized methods applied to fluids. Comput. Methods Appl. Mech. Engrg. 105(3), 395–403 (1993). https://doi.org/10.1016/0045-7825(93)90065-6 Franca, L.P., Madureira, A.L., Valentin, F.: Towards multiscale functions: enriching finite element spaces with local but not bubble-like functions. Comput. Methods Appl. Mech. Engrg. 194(27-29), 3006–3021 (2005). https://doi.org/10.1016/j.cma.2004.07.029 Harder, C., Madureira, A.L., Valentin, F.: A hybrid-mixed method for elasticity. ESAIM Math. Model. Numer. Anal. 50(2), 311–336 (2016). https://doi.org/10.1051/m2an/2015046 Harder, C., Paredes, D., Valentin, F.: A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coefficients. J. Comput. Phys. 245, 107–130 (2013). https://doi.org/10.1016/j.jcp.2013.03.019 Harder, C., Paredes, D., Valentin, F.: On a multiscale hybrid-mixed method for advective-reactive dominated problems with heterogenous coefficients. SIAM Multiscale Model. and Simul. 13(2), 491–518 (2015). https://doi.org/10.1137/130938499 Henning, P., Målqvist, A., Peterseim, D.: A localized orthogonal decomposition method for semi-linear elliptic problems. ESAIM Math. Model. Numer. Anal. 48(5), 1331–1349 (2014). https://doi.org/10.1051/m2an/2013141 Henning, P., Ohlberger, M.: A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete Contin. Dyn. Syst. Ser. S 9(5), 1393–1420 (2016). https://doi.org/10.3934/dcdss.2016056 Hou, T.Y., Wu, X.H.: A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997). https://doi.org/10.1006/jcph.1997.5682 Hou, T.Y., Wu, X.H., Cai, Z.: Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comp. 68(227), 913–943 (1999). https://doi.org/10.1090/S0025-5718-99-01077-7 Irisarri, D., Hauke, G.: A posteriori pointwise error computation for 2-D transport equations based on the variational multiscale method. Comput. Methods Appl. Mech. Engrg. 311, 648–670 (2016). https://doi.org/10.1016/j.cma.2016.09.001 John, V.: Residual a posteriori error estimates for two-level finite element methods for the Navier-Stokes equations. Appl. Numer. Math. 37(4), 503–518 (2001). https://doi.org/10.1016/S0168-9274(00)00058-1 John, V.: Finite Element Methods for Incompressible Flow Problems. Springer Series in Computational Mathematics, vol. 51. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45750-5 Ohlberger, M.: A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems. Multiscale Model Simul. 4(1), 88–114 (2005). https://doi.org/10.1137/040605229 Paredes, D., Valentin, F., Versieux, H.M.: On the robustness of multiscale hybrid-mixed methods. Math. Comp. 86(304), 525–548 (2017). https://doi.org/10.1090/mcom/3108 Pencheva, G.V., Vohralík, M., Wheeler, M.F., Wildey, T.: Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling. SIAM J. Numer. Anal. 51(1), 526–554 (2013). https://doi.org/10.1137/110839047 Pope, S.B.: Turbulent Flows. Cambridge University Press. (2000). https://doi.org/10.1017/CBO9780511840531675 Raviart, P.A., Thomas, J.M.: Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comp. 31(138), 391–413 (1977). https://doi.org/10.2307/2006423 Ren, C., Ma, Y.: Residual a posteriori error estimate of a new two-level method for steady Navier-Stokes equations. J. Syst. Sci. Complex. 19 (4), 478–490 (2006). https://doi.org/10.1007/s11424-006-0478-5 Tavener, S., Wildey, T.: Adjoint based a posteriori analysis of multiscale mortar discretizations with multinumerics. SIAM J. Sci. Comput. 35 (6), A2621–A2642 (2013). https://doi.org/10.1137/12089973X Xu, J.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33(5), 1759–1777 (1996). https://doi.org/10.1137/S0036142992232949 Zhang, Y.Z., Hou, Y.R., Wei, H.B.: Residual a posteriori error estimate of two level finite element method for natural convection problem. Gongcheng Shuxue Xuebao 32(1), 116–130 (2015)