Ultraconvergence of the patch recovery technique II. (English) Zbl 0936.65132
Summary: [For part I, see ibid. 85, No. 216, 1431-1437 (1996; Zbl 0853.65116).]
The ultraconvergence property of a gradient recovery technique proposed by O. C. Zienkiewicz and J. Z. Zhu [Int. J. Numer. Methods Eng. 33, No. 7, 1331-1364 (1992; Zbl 0769.73084)] is analyzed for the Laplace equation in the two-dimensional setting. Under the assumption that the pollution effect is not present or is properly controlled, it is shown that the convergence rate of the recovered gradient at an interior node is two orders higher than the optimal global convergence rate when even-order finite element spaces and local uniform rectangular meshes are used.
The ultraconvergence property of a gradient recovery technique proposed by O. C. Zienkiewicz and J. Z. Zhu [Int. J. Numer. Methods Eng. 33, No. 7, 1331-1364 (1992; Zbl 0769.73084)] is analyzed for the Laplace equation in the two-dimensional setting. Under the assumption that the pollution effect is not present or is properly controlled, it is shown that the convergence rate of the recovered gradient at an interior node is two orders higher than the optimal global convergence rate when even-order finite element spaces and local uniform rectangular meshes are used.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
Keywords:
patch recovery technique; ultraconvergence; gradient recovery technique; Laplace equation; finite elementReferences:
| [1] | Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937 – 958. · Zbl 0298.65071 |
| [2] | A. H. Schatz, I. H. Sloan, and L. B. Wahlbin, Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point, SIAM J. Numer. Anal. 33 (1996), no. 2, 505 – 521. · Zbl 0855.65115 · doi:10.1137/0733027 |
| [3] | A. H. Schatz and L. B. Wahlbin, Interior maximum-norm estimates for finite element methods. II, Math. Comp. 64 (1995), no. 211, 907 – 928. · Zbl 0826.65091 |
| [4] | Barna Szabó and Ivo Babuška, Finite element analysis, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1991. · Zbl 0792.73003 |
| [5] | Lars B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. · Zbl 0826.65092 |
| [6] | Zhimin Zhang, Ultraconvergence of the patch recovery technique, Math. Comp. 65 (1996), no. 216, 1431 – 1437. · Zbl 0853.65116 |
| [7] | Zhimin Zhang and Harold Dean Victory Jr., Mathematical analysis of Zienkiewicz-Zhu’s derivative patch recovery technique, Numer. Methods Partial Differential Equations 12 (1996), no. 4, 507 – 524. , https://doi.org/10.1002/(SICI)1098-2426(199607)12:43.0.CO;2-Q · Zbl 0858.65115 |
| [8] | O.C. Zienkiewicz and J.Z. Zhu, The superconvergence patch recovery and a posteriori error estimates. Part 1: The recovery technique, Internat. J. Numer. Meth. Eng. 33 (1992), 1331-1364. · Zbl 0769.73084 |
| [9] | Chao Ding Zhu and Qun Lin, Youxianyuan chaoshoulian lilun, Hunan Science and Technology Publishing House, Changsha, 1989 (Chinese). |
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