Nitsche, J. A.; Wheeler, Mary F. \(L_{\infty}\)-boundedness of the finite element Galerkin operator for parabolic problems. (English) Zbl 0533.65071 Numer. Funct. Anal. Optimization 4, 325-353 (1982). The initial-boundary value problem for the evolution equation (the heat equation) \(\dot u-\Delta u=f\) in \(\Omega\times(0,T]\), \(\Omega \subseteq R^ N\) (\(N\leq 3)\) with boundary conditions \(u=0\) on \(\partial \Omega \times(0,T]\) and \(u_{t=0}=u_ 0\) in \(\Omega\) is considered. The main result of this paper is: In \(N=1,2,3\) space dimensions and for finite elements of order 4 or higher, i.e. at least cubics, the mapping \(u\to u_ h\) (where h is the mesh size of the approximation) is bounded in \(L_{\infty}\): \[ \| u_ h\|_{L_{\infty}(0,T,L_{\infty}(\Omega))} \leq C\| u\|_{L_{\infty}(0,T,L_{\infty}(\Omega))}. \] Reviewer: J.Lovíšek Cited in 9 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35K05 Heat equation 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations Keywords:Galerkin method; uniform error bound; cubic elements; finite elements in weighted norms PDFBibTeX XMLCite \textit{J. A. Nitsche} and \textit{M. F. Wheeler}, Numer. Funct. Anal. Optim. 4, 325--353 (1982; Zbl 0533.65071) Full Text: DOI References: [1] Bramble J.H., SIAM J. Numer. Anal. 14 pp 218– (1977) · Zbl 0364.65084 · doi:10.1137/0714015 [2] Ciarlet P.G., Comput. Methods Appl. Mech. Engrg. 1 pp 217– (1972) · Zbl 0261.65079 · doi:10.1016/0045-7825(72)90006-0 [3] Dobrowolski M., Meth. Appl. Sci. 2 pp 221– (1980) · Zbl 0434.65088 · doi:10.1002/mma.1670020208 [4] Dobrowolsi M., SIAM J. Numer. Anal. 17 pp 663– (1980) · Zbl 0449.65077 · doi:10.1137/0717056 [5] Gilbarg, D. and Trudinger, N.S. 1977. ”Elliptic Partial Differential Equations of Second Order.”. Berlin: Springer Verlag. · Zbl 0361.35003 · doi:10.1007/978-3-642-96379-7 [6] Nitsche J., R.A.I.R.O. 12 pp 31– (1979) [7] Nitsche, J. 1981. Schauder estimates for finite element approximations on second order elliptic boundary value problems. Proceedings of the Special Year in Numerical Analysis, Lecture Notes No.20. 1981. Edited by: Babuska, I., Liu, T.P. and Osborn, J. pp.290–343. Univ. of Maryland. [8] Schatz A.H., Comm. Pure Appl. Math. 33 pp 265– (1980) · Zbl 0414.65066 · doi:10.1002/cpa.3160330305 [9] Thomee V., The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations pp 711– (1972) · doi:10.1016/B978-0-12-068650-6.50031-9 [10] Thomee V., Mathematical Aspects of Finite Elements in Partial Differential Equations pp 55– (1974) · doi:10.1016/B978-0-12-208350-1.50007-8 [11] Wahlbin, L.B. Dundee Biennial Conference on Numerical Analysis. June91981. Edited by: Watson, G.A. Vol. 912, pp.230–245. Berlin: Springer Verlag. Lecture Notes in Mathematics 23-26 [12] Wheeler M.F., SIAM J. Numer. Anal. 10 pp 908– (1973) · Zbl 0266.65074 · doi:10.1137/0710076 [13] Zlamal M., I. SIAM J. Numer. Anal., II. SIAM J. Numer. Anal. 10 pp 229, 347– (1973) 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.