×

A mass conserving mixed stress formulation for Stokes flow with weakly imposed stress symmetry. (English) Zbl 1439.35368


MSC:

35Q30 Navier-Stokes equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
76M10 Finite element methods applied to problems in fluid mechanics

Software:

NGSolve
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] D. N. Arnold, F. Brezzi, and J. Douglas, Jr., PEERS: A new mixed finite element for plane elasticity, Japan J. Appl. Math., 1 (1984), pp. 347-367. · Zbl 0633.73074
[2] D. N. Arnold, R. S. Falk, and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp., 76 (2007), pp. 1699-1723. · Zbl 1118.74046
[3] P. B. Bochev and M. D. Gunzburger, Least-squares methods for the velocity-pressure-stress formulation of the stokes equations, Comput. Methods Appl. Mech. Engrg., 126 (1995), pp. 267-287. · Zbl 1067.76562
[4] D. Boffi, F. Brezzi, and M. Fortin, Reduced symmetry elements in linear elasticity, Commun. Pure Appl. Anal., 8 (2009), pp. 95-121. · Zbl 1154.74041
[5] D. Boffi, F. Brezzi, and M. Fortin, Mixed Finite Element Methods and Applications, Springer, New York, 2013. · Zbl 1277.65092
[6] S. C. Brenner, Korn’s inequalities for piecewise \(H^1\) vector fields, Math. Comp., 73 (2004), pp. 1067-1087. · Zbl 1055.65118
[7] F. Brezzi, J. Douglas, Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), pp. 217-235. · Zbl 0599.65072
[8] Z. Cai, B. Lee, and P. Wang, Least-squares methods for incompressible Newtonian fluid flow: Linear stationary problems, SIAM J. Numer. Anal., 42 (2004), pp. 843-859. · Zbl 1159.76347
[9] B. Cockburn and J. Gopalakrishnan, A characterization of hybridized mixed methods for the Dirichlet problem, SIAM J. Numer. Anal., 42 (2004), pp. 283-301. · Zbl 1084.65113
[10] B. Cockburn, J. Gopalakrishnan, and J. Guzmán, A new elasticity element made for enforcing weak stress symmetry, Math. Comp., 79 (2010), pp. 1331-1349. · Zbl 1369.74078
[11] B. Cockburn, G. Kanschat, and D. Schötzau, A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comp., 74 (2005), pp. 1067-1095. · Zbl 1069.76029
[12] B. Cockburn, G. Kanschat, and D. Schötzau, A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, J. Sci. Comput., 31 (2007), pp. 61-73. · Zbl 1151.76527
[13] B. Cockburn and F.-J. Sayas, Divergence-conforming HDG methods for Stokes flows, Math. Comp., 83 (2014), pp. 1571-1598. · Zbl 1427.76125
[14] M. Farhloul, Mixed and nonconforming finite element methods for the Stokes problem, Can. Appl. Math. Q., 3 (1995). · Zbl 0846.35100
[15] M. Farhloul and M. Fortin, A new mixed finite element for the Stokes and elasticity problems, SIAM J. Numer. Anal., 30 (1993), pp. 971-990. · Zbl 0777.76051
[16] M. Farhloul and M. Fortin, Dual hybrid methods for the elasticity and the Stokes problems: A unified approach, Numer. Math., 76 (1997), pp. 419-440. · Zbl 0880.73064
[17] M. Farhloul and M. Fortin, Review and complements on mixed-hybrid finite element methods for fluid flows, J. Comput. Appl. Math., 140 (2002), pp. 301-313. · Zbl 1134.76383
[18] G. Fu, Y. Jin, and W. Qiu, Parameter-free superconvergent \(H\)(div)-conforming HDG methods for the Brinkman equations, IMA J. Numer. Anal., (2018), dry001.
[19] G. N. Gatica, L. F. Gatica, and A. Márquez, Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow, Numer. Math., 126 (2014), pp. 635-677. · Zbl 1426.74232
[20] G. N. Gatica, A. Márquez, and M. A. Sánchez, Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1064-1079. · Zbl 1227.76030
[21] G. N. Gatica, A. Márquez, and M. A. Sánchez, A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows, Comput. Methods Appl. Mech. Engrg., 200 (2011), pp. 1619-1636. · Zbl 1228.76084
[22] M. Giacomini, A. Karkoulias, R. Sevilla, and A. Huerta, A superconvergent HDG method for Stokes flow with strongly enforced symmetry of the stress tensor, J. Sci. Comput., 77 (2018), pp. 1679-1702. · Zbl 1404.76162
[23] J. Gopalakrishnan and J. Guzmán, A second elasticity element using the matrix bubble, IMA J. Numer. Anal., 32 (2012), pp. 352-372. · Zbl 1232.74101
[24] J. Gopalakrishnan, P. L. Lederer, and J. Schöberl, A mass conserving mixed stress formulation for the Stokes equations, IMA J. Numer. Anal., (2019), drz022, https://doi.org/10.1093/imanum/drz022.
[25] J. Guzmán, C.-W. Shu, and F. A Sequeira, \(H({div})\) conforming and DG methods for incompressible Euler’s equations, IMA J. Numer. Anal., (2016), drw054. · Zbl 1433.76085
[26] J. Könnö and R. Stenberg, Numerical computations with H(div)-finite elements for the Brinkman problem, Comput. Geosci., 16 (2012), pp. 139-158. · Zbl 1348.76100
[27] P. L. Lederer, A Mass Conserving Mixed Stress Method for Incompressible Flows, Ph.D. thesis, Technical University of Vienna, 2019.
[28] P. L. Lederer, C. Lehrenfeld, and J. Schöberl, Hybrid Discontinuous Galerkin methods with relaxed \(H{\rm(div)}\)-conformity for incompressible flows. Part I, SIAM J. Numer. Anal., 56 (2018), pp. 2070-2094. · Zbl 1402.35209
[29] P. L. Lederer, C. Lehrenfeld, and J. Schöberl, Hybrid Discontinuous Galerkin methods with relaxed \(H(div)\)-conformity for incompressible flows. Part II, ESAIM Math. Model. Numer. Anal., 53 (2019), pp. 503-522, https://doi.org/10.1051/m2an/2018054. · Zbl 1434.35058
[30] P. L. Lederer, A. Linke, C. Merdon, and J. Schöberl, Divergence-free reconstruction operators for pressure-robust Stokes discretizations with continuous pressure finite elements, SIAM J. Numer. Anal., 55 (2017), pp. 1291-1314. · Zbl 1457.65202
[31] P. L. Lederer and J. Schöberl, Polynomial robust stability analysis for \(H\)(div)-conforming finite elements for the Stokes equations, IMA J. Numer. Anal., 38 (2017), pp. 1832-1860, https://doi.org/10.1093/imanum/drx051. · Zbl 1462.65192
[32] C. Lehrenfeld and J. Schöberl, High order exactly divergence-free hybrid discontinuous Galerkin methods for unsteady incompressible flows, Comput. Methods Appl. Mech. Engrg., 307 (2016), pp. 339-361.
[33] P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, in Mathematical Aspects of Finite Element Methods, Lecture Notes in Math. 606, Springer, Berlin, 1977, pp. 292-315. · Zbl 0362.65089
[34] J. Schöberl, NETGEN An advancing front \(2\) D/\(3\) D-mesh generator based on abstract rules, Comput. Vis. Sci., 1 (1997), pp. 41-52. · Zbl 0883.68130
[35] J. Schöberl, C++11 Implementation of Finite Elements in NGSolve, Tech. Report ASC-2014-30, Institute for Analysis and Scientific Computing, 2014.
[36] R. Stenberg, A family of mixed finite elements for the elasticity problem, Numer. Math., 53 (1988), pp. 513-538. · Zbl 0632.73063
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.