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Analysis of mortar-type \(Q_1^{\mathrm{rot}}/Q_0\) element and multigrid methods for the incompressible Stokes problem. (English) Zbl 1112.76045

Summary: We propose a mortar-type \(Q_1^{\mathrm{rot}}/Q_0\) element method for incompressible Stokes problem, and obtain the optimal error estimate. Meanwhile, a \(\mathcal W\)-cycle multigrid method is given for solving the discrete problem, and the optimal convergence of the multigrid method is proved. Finally, numerical experiments are presented to confirm our theoretical results.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
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[1] Adams, R. A., Sobolev Space (1975), Academic Press: Academic Press New York
[2] Arbogast, T.; Chen, Z., On the implementation of mixed methods as nonconforming methods for second order elliptic problems, Math. Comp., 64, 943-971 (1995) · Zbl 0829.65127
[3] Belgacem, F. B., The mixed mortar finite element method for the incompressible Stokes problem: Convergence analysis, SIAM J. Numer. Anal., 37, 1085-1100 (2000) · Zbl 0959.65126
[4] Bernardi, C.; Maday, Y.; Patera, A., A new nonconforming approach to domain decomposition: The mortar element method, (Brezis, H.; Lions, J. L., Nonlinear Differential Equations and Their Applications. Nonlinear Differential Equations and Their Applications, College de France Seminar, vol. XI (1994), Pitman: Pitman Boston), 13-51 · Zbl 0797.65094
[5] Boland, J. M.; Nicolaides, R. A., Stability of finite elements under divergence constraints, SIAM J. Numer. Anal., 20, 722-731 (1983) · Zbl 0521.76027
[6] Braess, D.; Dahmen, W.; Wieners, C., A multigrid algorithm for the mortar finite element method, SIAM J. Numer. Anal., 37, 48-69 (1999) · Zbl 0942.65139
[7] Braess, D.; Dryja, M.; Hackbusch, W., A multigrid method for nonconforming FE-discretizations with application to non-matching grids, Computing, 63, 1-25 (1999) · Zbl 0931.65120
[8] Braess, D.; Sarazin, R., An efficient smoother for the Stokes problem, Appl. Numer. Math., 23, 3-20 (1997) · Zbl 0874.65095
[9] Brenner, S. C.; Scott, L. R., The Mathematical Theory of Finite Element Methods (1994), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0804.65101
[10] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO Anal. Numer., 8, 129-151 (1974) · Zbl 0338.90047
[11] Chen, J.; Xu, X., The mortar element method for rotated \(Q_1\) element, J. Comput. Math., 20, 313-324 (2002) · Zbl 1015.65066
[12] Chen, Z.; Oswald, P., Multigrid and multilevel methods for nonconforming \(Q_1\) element, Math. Comp., 67, 667-693 (1998) · Zbl 0893.65060
[13] Cialet, P. G., The Finite Element Methods for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam
[14] Girault, V.; Raviart, P. A., Finite Element Methods for Navier-Stokes Equations (1986), Springer-Verlag: Springer-Verlag Berlin, New York, Heidelberg · Zbl 0396.65070
[15] John, V.; Tobiska, L., A coupled multigrid method for nonconforming finite element discretizations of the 2D-Stokes equation, Computing, 64, 307-321 (2000) · Zbl 0963.65126
[16] Kloucek, P.; Li, B.; Luskin, M., Analysis of a class of non-conforming finite elements for crystalline microstructure, Math. Comp., 65, 1111-1135 (1996) · Zbl 0903.65081
[17] Rannacher, R.; Turek, S., Simple nonconforming quadrilateral Stokes element, Numer. Meth. Part. Diff. Eq., 8, 97-111 (1992) · Zbl 0742.76051
[18] Shi, Z.; Xu, X., A V-cycle multigrid for quadrilateral rotated \(Q_1\) element with numerical integration, J. Comput. Math., 21, 545-554 (2003) · Zbl 1036.65110
[19] Vefürth, R., A multilevel algorithm for mixed problems, SIAM J. Numer. Anal., 21, 264-271 (1984) · Zbl 0534.65065
[20] Xu, X.; Chen, J., Multigrid for mortar element method for \(P_1\) nonconforming element, Numer. Math., 88, 381-398 (2001) · Zbl 0989.65141
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