Approximation of viscosity solutions of elliptic partial differential equations on minimal grids
Estimates in terms of the ellipticity are obtained for the minimal size \(N_A\) of an equally spaced grid used in the approximation of solutions to elliptic partial differential equations. Because of the equal spacing, the results involve the theory of Diophantine approximation and particularly continued fractions. A general upper bound for the minimal size is established for the strictly elliptic problem by approximating orthogonal bases in \(\mathbb{R}^n\) by orthogonal bases with integer coordinates [this result is proved separately in an Appendix by \textit{W. M. Schmidt}, ibid. 72, No. 1, 117-122 (1995; reviewed below)]. In the two-dimensional case, the minimal size is infinite or \(\max\{n, m\}\) according as the ratio of the diagonal terms of the matrix \(A\) is irrational or \(n/m\) (\(m\), \(n\) coprime). Bounds for \(N_A\) are obtained in the two-dimensional case when the matrix \(A\) of the differential equation is of the form \[ A= (\begin{smallmatrix} 1\\ \sqrt\alpha\end{smallmatrix} \begin{smallmatrix} \sqrt\alpha\\ \alpha+ \varepsilon\end{smallmatrix}), \] \(\alpha, \varepsilon> 0\), \(\varepsilon\) small and \(\alpha\neq 1\). When \(\sqrt\alpha\) is irrational, \(N_A\leq C/\varepsilon\) and when \(\sqrt\alpha\) is badly approximable, \(N_A\leq C/\sqrt\varepsilon\). The last estimate cannot be improved in general although when \(\sqrt\alpha\) is badly approximable, \(N_A\geq C\varepsilon^{- 1/4}\), an estimate which also holds for irrational \(\sqrt\alpha\) for an infinite sequence of \(\varepsilon\).
scientific article; zbMATH DE number 839149