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Optimal-order finite difference approximation of generalized solutions to the biharmonic equation in a cube. (English) Zbl 1471.65172

The authors proved an optimal-order error bound for finite difference approximation of the first boundary-value problem for the biharmonic equation in \(\Omega=(0,1)^n\), with \(n\in \{2,\dots,7\}\), whose generalized solution belongs to the Sobolev space \(H^s(\Omega)\cap H_0^2(\Omega)\) for \(\frac{1}{2}\max (5,n)<s\leq 4.\) One of main results improves results from [B. S. Jovanović and the third author, Analysis of finite difference schemes. For linear partial differential equations with generalized solutions. London: Springer (2014; Zbl 1335.65071)] and [I. P. Gavrilyuk et al., Sov. Math. 27, No. 2, 13–21 (1983; Zbl 0566.65071)]. In order to compare the finite difference approximation with the original problem the authors use a different, carefully chosen extension of the generalized solution \(u\) from \(\Omega\) to \(\mathbb{R}^N \setminus \Omega\) which is the main novelty of the paper. The results of the paper could also be of interest in statistical mechanics and probability.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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References:

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