Superconvergence of solution derivatives for the Shortley–Weller difference approximation of Poisson’s equation. I: Smoothness problems. (English) Zbl 1030.65108
This paper explores the superconvergence of derivatives for the Shortley-Weller approximation [cf. N. Matsunuga and T. Yamamoto, ibid. 116, 263-273 (2000; Zbl 0952.65082)]. The finite difference method (FDM) using the Shortley-Weller approximation can be viewed as a special kind of the finite element methods (FEMs) using the piecewise bilinear and linear functions, and involving some integration approximation. When \(u \in C^{3}( {\overline {S}})\) (i.e., \(u \in C^{3,0}( {\overline {S}}))\) and \(f \in C^{2}( {\overline {S}}), \)the superconvergence rate \(O( {h^{2}})\) of solution derivatives in discrete \(H^{1}\) norms by the FDM is derived for rectangular difference grids, where \(h\) is the maximal mesh length of difference grids used, and the difference grids are not confined to be quasiuniform.
Three kinds of FEMs are considered, and display a deeper relation between the FDM and the FEM. Based on numerical results, the Shortley-Weller approximation is the best. The superconvergence analysis of this paper can be applied to all of them. Not only can the FEM analysis be employed to the Shortley-Weller approximation, but also the traditional FDM analysis using the maximum principle to the bilinear FEM with uniform rectangles.
Numerical experiments for the scheme of J. H. Bramble and B. E. Hubbard [Numer. Math. 4, 313-327 (1962; Zbl 0135.18102)] are carried out, and the maximal nodal errors are \(O( {h^{4}})\) numerically. By means of the a posteriori interpolant, the derivatives of order \(k\) have the errors \(O( {h^{4 - k}})\), \(k = 1,2,3\). For the uniform rectangular grid with \(h_{i} = k_{j} = h,\) the Bramble-Hubbard scheme is strongly recommended due to high convergence rates.
Three kinds of FEMs are considered, and display a deeper relation between the FDM and the FEM. Based on numerical results, the Shortley-Weller approximation is the best. The superconvergence analysis of this paper can be applied to all of them. Not only can the FEM analysis be employed to the Shortley-Weller approximation, but also the traditional FDM analysis using the maximum principle to the bilinear FEM with uniform rectangles.
Numerical experiments for the scheme of J. H. Bramble and B. E. Hubbard [Numer. Math. 4, 313-327 (1962; Zbl 0135.18102)] are carried out, and the maximal nodal errors are \(O( {h^{4}})\) numerically. By means of the a posteriori interpolant, the derivatives of order \(k\) have the errors \(O( {h^{4 - k}})\), \(k = 1,2,3\). For the uniform rectangular grid with \(h_{i} = k_{j} = h,\) the Bramble-Hubbard scheme is strongly recommended due to high convergence rates.
Reviewer: Leonid B.Chubarov (Novosibirsk)
MSC:
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
Keywords:
Superconvergence; Shortley-Weller difference approximation; Poisson’s equation; finite difference method; finite element methods; comparison of methods; convergence; numerical resultsReferences:
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