×

Full-order convergence of a mixed finite element method for fourth-order elliptic equations. (English) Zbl 0923.65074

The paper deals with the mixed finite element method for a fourth-order elliptic equation on a rectangular mesh. By using a special interpolation operator and an elaborate element analysis, the author improves the classical error estimates to full order.
Reviewer: P.Burda (Praha)

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J40 Boundary value problems for higher-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Babuska, I., The finite element method with Lagrangian multipliers, Numer. Math., 20, 179-192 (1973) · Zbl 0258.65108
[2] Babuska, I.; Osborn, J.; Pitkäranta, J., Analysis of mixed methods using mesh-dependent norms, Math. Comp., 35, 1039-1079 (1980) · Zbl 0472.65083
[3] Babuska, I.; Oden, J. T., Special Issue on \(p hp \), Comput. Methods Appl. Mech. Engrg., 133 (1996)
[4] Babuska, I.; Suri, M., The \(php\), SIAM Rev., 36, 578-632 (1994) · Zbl 0813.65118
[5] Brezzi, F.; Douglas, J.; Fortin, M.; Marini, L. D., Efficient rectangular mixed finite elements in two and three space variables, RAIRO Math. Modelling Numer. Anal., 21, 581-604 (1987) · Zbl 0689.65065
[6] Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0788.73002
[7] Brezzi, F.; Raviart, P. A., Mixed finite element methods for 4th-order elliptic equations, (Miller, J., Topics in Numerical Analysis III (1978), Academic Press: Academic Press San Diego) · Zbl 0434.65085
[8] Ciarlet, P. G., The Finite Element Methods for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0445.73043
[9] Ciarlet, P. G.; Raviart, P. A., A mixed finite element method for the biharmonic equation, (de Boor, C., Mathematical Aspects of Finite Elements in Partial Differential Equations (1974), Academic Press: Academic Press San Diego) · Zbl 0337.65058
[10] Demkowicz, L.; Oden, J. T.; Rachowicz, W.; Hardy, O., Toward a universal \(hp\), Comput. Methods Appl. Mech. Engrg., 77, 79-112 (1989) · Zbl 0723.73074
[11] Douglas, J.; Wang, J., Superconvergence of mixed finite element methods on rectangular domains, Calcolo, 26, 121-134 (1989) · Zbl 0714.65084
[12] Duran, R., Superconvergence for rectangular mixed finite elements, Numer. Math., 58, 287-298 (1990) · Zbl 0691.65076
[13] Ewing, R. E.; Lazarov, R. D.; Wang, J., Superconvergence of the velocity along the Gauss lines in mixed finite element methods, SIAM J. Numer. Anal., 28, 1015-1029 (1991) · Zbl 0733.65065
[14] Ewing, R. E.; Lazarov, R. D., Superconvergence of the mixed finite element approximations of parabolic problems using rectangular finite elements, East-West J. Numer. Math., 1, 199-212 (1993) · Zbl 0835.65105
[15] Falk, R. S.; Osborn, J. E., Error estimates for mixed methods, RAIRO Numer. Anal., 14, 249-277 (1981) · Zbl 0467.65062
[16] Girault, V.; Raviart, P. A., Finite Element Methods for Navier-Stokes Equations-Theory and Algorithms (1986), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0396.65070
[17] Lin, Q.; Li, J.; Zhou, A., A rectangular test for Ciarlet-Raviart and Hermann-Miyoshi schemes, Proc. of Systems Science and Systems Engineering (1991), Culture Publishing Co: Culture Publishing Co Hong Kong
[18] Lin, Q.; Yan, N., The Construction and Analysis of High Accurate Finite Element Methods (1996), Hebei Univ. Press: Hebei Univ. Press Hebei
[19] Nakata, M.; Weiser, A.; Wheeler, M. F., Some superconvergence results for mixed finite element methods for elliptic problems on rectangular domains, (Whiteman, J. R., The Mathematics of Finite Elements and Applications (1985), Academic Press: Academic Press London) · Zbl 0583.65078
[20] Oden, J. T., Some contributions to the mathematical theory of mixed finite element methods, Theory and Practice in Finite Element Structure Analysis (1973), Univ. of Tokyo Press: Univ. of Tokyo Press Tokyo · Zbl 0374.65060
[21] Oden, J. T.; Carey, G. F., Finite Elements: Mathematical Aspects (1983), Prentice-Hall: Prentice-Hall Englewood Cliffs · Zbl 0496.65055
[22] Rachowicz, W.; Oden, J. T.; Demkowicz, L., Toward a universal \(hphp\), Comput. Methods Appl. Mech. Engrg., 77, 181-212 (1989) · Zbl 0723.73076
[23] Scholz, R., A mixed method for 4th order problems using linear finite elements, RAIRO Numer. Anal., 12, 85-90 (1978) · Zbl 0382.65059
[24] Wahlbin, L. B., Superconvergence in Galerkin Finite Element Methods. Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, 1605 (1995), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0826.65092
[25] Zhang, Z.; Zhang, S., Derivative superconvergence of rectangular finite elements for the Reissner-Mindlin plate, Comput. Methods Appl. Mech. Engrg., 134, 1-16 (1996) · Zbl 0891.73071
[26] Zhou, A.; Li, J., The full approximation accuracy for the stream function-vorticity pressure method, Numer. Math., 68, 427-435 (1994) · Zbl 0823.65110
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.