Pehlivanov, A. I.; Carey, G. F.; Lazarov, R. D.; Shen, Y. Convergence analysis of least-squares mixed finite elements. (English) Zbl 0790.65079 Computing 51, No. 2, 111-123 (1993). The paper deals with theoretical error estimates for the least squares finite element method applied to a system of first order ordinary differential equations. The estimates are consistent with the numerical studies of G. F. Carey and Y. Shen [Commun. Appl. Numer. Methods 5, No. 7, 427-434 (1989; Zbl 0684.65083)] for the same system. It is proved that the least squares method is not subject to the Ladyzhenskaya-Babuska-Brezzi condition. Some questions are still to be resolved and it is also indicated that the results may be extended to higher dimensions. Reviewer: V.Subba Rao (Madras) Cited in 13 Documents MSC: 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65L70 Error bounds for numerical methods for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems Keywords:error estimates; least squares finite element method; system of first order ordinary differential equations; Ladyzhenskaya-Babuska-Brezzi condition Citations:Zbl 0684.65083 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Bramble, J. H., Hilbert, S. R.: Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal.7, 113–124 (1970). · Zbl 0201.07803 · doi:10.1137/0707006 [2] Brezzi, F.: On the existence, uniqueness and approximation of sadle point problems arising from lagrangian multipliers. RAIRO Sér. Anal. Numér.8, no. R-2, 129–151 (1974). · Zbl 0338.90047 [3] Carey, G. F., Humphrey, D., Wheeler, M. F.: Galerkin and collocation-Galerkin methods with superconvergence and optimal fluxes. Int. J. Numer. Meth. Engrg.17, 939–950 (1981). · Zbl 0461.65064 · doi:10.1002/nme.1620170610 [4] Carey, G. F., Oden, J. T.: Finite elements: A second course. Englewood Cliffs, N.J.: Prentice Hall 1983. · Zbl 0515.65075 [5] Carey, G. F., Shen, Y.: Convergence studies of least squares finite elements for first-order systems. Comm. Appl. Numer. Meth.5, 427–434 (1989). · Zbl 0684.65083 · doi:10.1002/cnm.1630050702 [6] Ciarlet, P. G.: The finite element method for elliptic problems. Amsterdam, New York, Oxford: North-Holland 1978. · Zbl 0383.65058 [7] Douglas, J. Jr., Dupont, T.: Galerkin approximations for the two points boundary problem using continuous, piecewise polynomial spaces. Numer. Math.22, 99–109 (1974). · Zbl 0331.65051 · doi:10.1007/BF01436724 [8] Dupont, T.: A unified theory of superconvergence for Galerkin methods for two-point boundary value problems. SIAM J. Numer. Anal.13, 362–368 (1976). · Zbl 0332.65050 · doi:10.1137/0713032 [9] Grisvard, P.: Elliptic problems in nonsmooth domains. London: Pitman 1985. · Zbl 0695.35060 [10] Jiang B. N., Chang, C. L.: Least-squares finite elements for the stokes problem. Comp. Meth. Appl. Mech. Engrng.78, 297–317 (1990). · Zbl 0706.76033 · doi:10.1016/0045-7825(90)90003-5 [11] Strang, G., Fix, G.: An analysis of the finite element method. Englewood Cliffs, N.J.: Prentice Hall 1973 · Zbl 0356.65096 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.