Ferreira, J. A.; Grigorieff, R. D. Supraconvergence and supercloseness of a scheme for elliptic equations on nonuniform grids. (English) Zbl 1102.65109 Numer. Funct. Anal. Optimization 27, No. 5-6, 539-564 (2006). Elliptic equations of second order are solved here by a finite difference method on anisotropic rectangular meshes. The finite difference method is considered as equivalent to a finite element method with a quadrature rule. Error estimates are derived via superconvergence. The domain is assumed to be polygonal, but also \(H^3\)-regularity is supposed which can be verified only for boundaries with a smoothness that is not encountered with polygonal domains. In a previous paper, even \(H^4\)-regularity was assumed, and a figure in the paper contains even a domain with a reentrant corner. Here we have even not \(H^2\)-regularity in the generic case. Reviewer: Dietrich Braess (Bochum) Cited in 19 Documents MSC: 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N06 Finite difference methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs Keywords:superconvergence; finite differences; finite element method; stability; supercloseness of gradient; second order elliptic equation; error estimates PDFBibTeX XMLCite \textit{J. A. Ferreira} and \textit{R. D. Grigorieff}, Numer. Funct. Anal. Optim. 27, No. 5--6, 539--564 (2006; Zbl 1102.65109) Full Text: DOI Link References: [1] DOI: 10.1093/imanum/dri018 · Zbl 1087.65070 · doi:10.1093/imanum/dri018 [2] Bojović D., CMAM 1 pp 213– (2001) [3] DOI: 10.1016/S0168-9274(98)00048-8 · Zbl 0929.65093 · doi:10.1016/S0168-9274(98)00048-8 [4] DOI: 10.1016/0168-9274(88)90016-5 · Zbl 0651.65086 · doi:10.1016/0168-9274(88)90016-5 [5] DOI: 10.1007/BF01385511 · Zbl 0762.65077 · doi:10.1007/BF01385511 [6] DOI: 10.2307/2938671 · Zbl 0727.65011 · doi:10.2307/2938671 [7] DOI: 10.1002/mana.19881350110 · Zbl 0654.65061 · doi:10.1002/mana.19881350110 [8] Hlaváĉek I., Apl. Mat. 32 pp 131– (1987) [9] Hlaváĉek I., Apl. Mat. 41 pp 241– (1996) [10] DOI: 10.1017/S0334270000004495 · Zbl 0583.65058 · doi:10.1017/S0334270000004495 [11] Jovanović B.S., CMAM 4 pp 48– (2004) · Zbl 1058.65096 · doi:10.2478/cmam-2004-0011 [12] B.S. Jovanović ( 1993 ). The finite difference method for boundary-value problems with weak solutions . Posebna Izdanja, 16, report . Matematički Institut u Beogradu , Belgrade . · Zbl 0801.65101 [13] Jovanović B.S., Mat. Vesnik 38 pp 131– (1986) [14] Jovanović B.S., Zh. Vychisl. Mat. Mat. Fiz. 39 pp 61– (1999) [15] DOI: 10.1093/imanum/7.3.301 · Zbl 0636.65096 · doi:10.1093/imanum/7.3.301 [16] DOI: 10.1090/S0025-5718-1986-0856701-5 · doi:10.1090/S0025-5718-1986-0856701-5 [17] K[rcirc]íĉek M., Numer. Math. 45 pp 105– (1987) [18] DOI: 10.1016/0377-0427(87)90018-5 · Zbl 0602.65084 · doi:10.1016/0377-0427(87)90018-5 [19] K[rcirc]íĉek M., Finite Element Methods pp 315– (1998) [20] DOI: 10.1007/BF01410107 · Zbl 0525.65069 · doi:10.1007/BF01410107 [21] Levermore C.D., Computational Techniques and Applications CTAC-87 (Sydney, 1987) pp 417– (1987) [22] DOI: 10.1093/imanum/5.4.407 · Zbl 0584.65067 · doi:10.1093/imanum/5.4.407 [23] DOI: 10.1090/S0025-5718-1986-0856700-3 · Zbl 0635.65093 · doi:10.1090/S0025-5718-1986-0856700-3 [24] M.A. Marletta ( 1988 ). Supraconvergence of Discretization Methods on Nonuniform Meshes . M.Sc. Thesis , Oxford University . [25] Nečas J., Les Méthodes Directes En Théorie Des Équations Elliptiques (1967) [26] DOI: 10.1016/0041-5553(69)90159-1 · doi:10.1016/0041-5553(69)90159-1 [27] Samarskij A.A., Theorie der Differenzenverfahren (1984) · Zbl 0543.65067 [28] DOI: 10.2307/2008127 · Zbl 0586.65064 · doi:10.2307/2008127 [29] Vabishchevich P.N., Comput. Math. Math. Phys. 38 pp 399– (1998) [30] Wahlbin L.B., Superconvergence in Galerkin Finite Element Methods (1995) · Zbl 0826.65092 · doi:10.1007/BFb0096835 [31] J.R. Whiteman and G. Goodsell ( 1987 ). Some gradient superconvergence results in the finite element method. In:Numerical Analysis and Parallel Processing( P.R. Turner , ed.), Lect. Notes in Math. 1397 . Springer , Berlin , pp. 182 – 260 . · Zbl 0674.65076 [32] Zlotnik A., CMAM 2 pp 295– (2002) · Zbl 1015.65059 · doi:10.2478/cmam-2002-0018 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.