×

Divergence-conforming discontinuous Galerkin finite elements for Stokes eigenvalue problems. (English) Zbl 1455.65204

The authors consider a divergence-conforming discontinuous Galerkin finite element method for Stokes eigenvalue problems. A priori error estimates for the eigenvalue and eigenfunction are derived. Moreover, reliable and efficient a posteriori error estimates of residual type are proved. Numerical results confirm their efficiency.
Reviewer: Wei Gong (Beijing)

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs

Software:

ARPACK; deal.ii; Amandus
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ainsworth, M.; Oden, Jt, A Posteriori Error Estimation in Finite Element Analysis (Pure and Applied Mathematics) (2000), New York: Wiley, New York · Zbl 1008.65076
[2] Armentano, Mg; Moreno, V., A posteriori error estimates of stabilized low-order mixed finite elements for the Stokes eigenvalue problem, J. Comput. Appl. Math., 269, 132-149 (2014) · Zbl 1294.65102 · doi:10.1016/j.cam.2014.03.027
[3] Arnold, Dn, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19, 4, 742-760 (1982) · Zbl 0482.65060 · doi:10.1137/0719052
[4] Babuška, I., Osborn, J.: Eigenvalue problems. In: Handbook of Numerical Analysis, Vol. II, Handb. Numer. Anal., II, pp. 641-787. North-Holland, Amsterdam (1991) · Zbl 0875.65087
[5] Bangerth, W.; Davydov, D.; Heister, T.; Heltai, L.; Kanschat, G.; Kronbichler, M.; Maier, M.; Turcksin, B.; Wells, D., The deal.II library, version 8.4, J. Numer. Math., 24, 3, 135-141 (2016) · Zbl 1348.65187 · doi:10.1515/jnma-2016-1045
[6] Bjørstad, Pe; Tjøstheim, Bp, High precision solutions of two fourth order eigenvalue problems, Computing, 63, 2, 97-107 (1999) · Zbl 0940.65119 · doi:10.1007/s006070050053
[7] Boffi, D., Finite element approximation of eigenvalue problems, Acta Numer., 19, 1-120 (2010) · Zbl 1242.65110 · doi:10.1017/S0962492910000012
[8] Boffi, D.; Gallistl, D.; Gardini, F.; Gastaldi, L., Optimal convergence of adaptive FEM for eigenvalue clusters in mixed form, Math. Comput., 86, 307, 2213-2237 (2017) · Zbl 1364.65234 · doi:10.1090/mcom/3212
[9] Boyer, F.; Fabrie, P., Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models (Applied Mathematical Sciences ) (2013), New York: Springer, New York · Zbl 1286.76005
[10] Brenner, Susanne C.; Monk, Peter; Sun, Jiguang, C 0 Interior Penalty Galerkin Method for Biharmonic Eigenvalue Problems, Lecture Notes in Computational Science and Engineering, 3-15 (2015), Cham: Springer International Publishing, Cham · Zbl 1349.65439
[11] Brezzi, F.; Douglas, J.; Marini, Ld, Two families of mixed finite elements for second order elliptic problems, Numerische Mathematik, 47, 2, 217-235 (1985) · Zbl 0599.65072 · doi:10.1007/BF01389710
[12] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics (1991), New York: Springer, New York · Zbl 0788.73002
[13] Cockburn, B.; Kanschat, G.; Schötzau, D., A locally conservative LDG method for the incompressible Navier-Stokes equations, Math. Comput., 74, 251, 1067-1095 (2005) · Zbl 1069.76029 · doi:10.1090/S0025-5718-04-01718-1
[14] Cockburn, B.; Kanschat, G.; Schötzau, D., A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations, J. Sci. Comput., 31, 1-2, 61-73 (2007) · Zbl 1151.76527 · doi:10.1007/s10915-006-9107-7
[15] Dörfler, W., A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal., 33, 3, 1106-1124 (1996) · Zbl 0854.65090 · doi:10.1137/0733054
[16] Gedicke, J.; Khan, A., Arnold-Winther mixed finite elements for Stokes eigenvalue problems, SIAM J. Sci. Comput., 40, 5, A3449-A3469 (2018) · Zbl 1407.65266 · doi:10.1137/17M1162032
[17] Girault, V.; Kanschat, G.; Rivière, B., Error analysis for a monolithic discretization of coupled Darcy and Stokes problems, J. Numer. Math., 22, 2, 109-142 (2014) · Zbl 1302.76101 · doi:10.1515/jnma-2014-0005
[18] Girault, V.; Raviart, Pa, Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics (1986), Berlin: Springer, Berlin
[19] Han, J.; Zhang, Z.; Yang, Y., A new adaptive mixed finite element method based on residual type a posterior error estimates for the Stokes eigenvalue problem, Numer. Methods Partial Differ. Equ., 31, 1, 31-53 (2015) · Zbl 1338.65237 · doi:10.1002/num.21891
[20] Houston, P.; Schötzau, D.; Wihler, Tp, Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem, J. Sci. Comput., 22, 23, 347-370 (2005) · Zbl 1065.76139
[21] Huang, P., Lower and upper bounds of Stokes eigenvalue problem based on stabilized finite element methods, Calcolo, 52, 1, 109-121 (2015) · Zbl 1317.65230 · doi:10.1007/s10092-014-0110-3
[22] Huang, P., He, Y., Feng, X.: Numerical investigations on several stabilized finite element methods for the Stokes eigenvalue problem. Math. Probl. Eng. p. Art. ID 745908, 14 (2011) · Zbl 1235.74286
[23] John, V.; Linke, A.; Merdon, C.; Neilan, M.; Rebholz, Lg, On the divergence constraint in mixed finite element methods for incompressible flows, SIAM Rev., 59, 3, 492-544 (2017) · Zbl 1426.76275 · doi:10.1137/15M1047696
[24] Kanschat, G.: Amandus. https://bitbucket.org/guidokanschat/amandus. A simple experimentation suite built on the dealii library. Accessed 16 Sept 2019
[25] Kanschat, G.; Mao, Y., Multigrid methods for \(H^{\text{div}} \)-conforming discontinuous Galerkin methods for the Stokes equations, J. Numer. Math., 23, 1, 51-66 (2015) · Zbl 1330.76071 · doi:10.1515/jnma-2015-0005
[26] Kanschat, G.; Schötzau, D., Energy norm a posteriori error estimation for divergence-free discontinuous Galerkin approximations of the Navier-Stokes equations, Int. J. Numer. Methods Fluids, 57, 9, 1093-1113 (2008) · Zbl 1140.76020 · doi:10.1002/fld.1795
[27] Kanschat, G.; Sharma, N., Divergence-conforming discontinuous Galerkin methods and \(C^0\) interior penalty methods, SIAM J. Numer. Anal., 52, 4, 1822-1842 (2014) · Zbl 1298.76117 · doi:10.1137/120902975
[28] Kellogg, Rb; Osborn, Je, A regularity result for the Stokes problem in a convex polygon, J. Funct. Anal., 21, 4, 397-431 (1976) · Zbl 0317.35037 · doi:10.1016/0022-1236(76)90035-5
[29] Lederer, Pl; Merdon, C.; Schöberl, J., Refined a posteriori error estimation for classical and pressure-robust Stokes finite element methods, Numer. Math., 142, 3, 713-748 (2019) · Zbl 1419.65097 · doi:10.1007/s00211-019-01049-3
[30] Lehoucq, R.; Sorensen, D.; Yang, C., ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods (1998), Philadelphia: SIAM, Philadelphia · Zbl 0901.65021
[31] Liu, H.; Gong, W.; Wang, S.; Yan, N., Superconvergence and a posteriori error estimates for the Stokes eigenvalue problems, BIT, 53, 3, 665-687 (2013) · Zbl 1278.65171 · doi:10.1007/s10543-013-0422-8
[32] Lovadina, C.; Lyly, M.; Stenberg, R., A posteriori estimates for the Stokes eigenvalue problem, Numer. Methods Partial Differ. Equ., 25, 1, 244-257 (2009) · Zbl 1169.65109 · doi:10.1002/num.20342
[33] Meddahi, S.; Mora, D.; Rodríguez, R., A finite element analysis of a pseudostress formulation for the Stokes eigenvalue problem, IMA J. Numer. Anal., 35, 2, 749-766 (2015) · Zbl 1416.65426 · doi:10.1093/imanum/dru006
[34] Mercier, B.; Osborn, J.; Rappaz, J.; Raviart, Pa, Eigenvalue approximation by mixed and hybrid methods, Math. Comput., 36, 154, 427-453 (1981) · Zbl 0472.65080 · doi:10.1090/S0025-5718-1981-0606505-9
[35] Schötzau, D.; Schwab, C.; Toselli, A., Mixed \(hp\)-DGFEM for incompressible flows, SIAM J. Numer. Anal., 40, 6, 2171-2194 (2002) · Zbl 1055.76032 · doi:10.1137/S0036142901399124
[36] Scott, Lr; Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comput., 54, 190, 483-493 (1990) · Zbl 0696.65007 · doi:10.1090/S0025-5718-1990-1011446-7
[37] Sun, J.; Zhou, A., Finite Element Methods for Eigenvalue Problems. Monographs and Research Notes in Mathematics (2017), Boca Raton: CRC Press, Boca Raton · Zbl 1351.65085
[38] Toselli, A., \(hp\) discontinuous Galerkin approximations for the Stokes problem, Math. Models Methods Appl. Sci., 12, 11, 1565-1597 (2002) · Zbl 1041.76045 · doi:10.1142/S0218202502002240
[39] Verfürth, R., A posteriori error estimation and adaptive mesh-refinement techniques, Journal of Computational and Applied Mathematics, 50, 1-3, 67-83 (1994) · Zbl 0811.65089 · doi:10.1016/0377-0427(94)90290-9
[40] Verfürth, R., A Review of a posteriori Error Estimation and Adaptive Mesh-refinement Techniques (1996), Leipzig: Teubner, Leipzig · Zbl 0853.65108
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.