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On the regularity of difference schemes. II: Regularity estimates for linear and nonlinear problems. (English) Zbl 0535.65071
[For the first part see ibid. 19, 71-95 (1981; Zbl 0462.65058).]
Let a boundary value problem for a second order elliptic equation be written in the form \(Lu=f\) and let \(L_ hu_ hh=f_ h\) be its discrete analogue. The author analyzes the important problem of getting a priori estimates for \(L_ h\!^{-1}\) which can be regarded as discrete analogue of the coercitivity estimates \[ (1)\quad \| L^{- 1}\|_{H^ s\to H^{s+2}}\leq C \] with \(H^ s\) being either the Sobolev space \(W^ s\!_ 2\) or the Hölder space \(C^ s\). He gives an interesting theorem which connects the \(s_ 1\)-regularity of the discrete problems with their \(s_ 0\)-regularity \((s_ 0<s_ 1)\) and estimates (1) for the differential case with \(s\in [s_ 0,s_ 1]\). This theorem enables him to get strong results about the s-regularity of a few difference schemes for the Poisson equation in a general region \(\Omega \subset R^ 2\) with a smooth boundary. Some generalizations for nonlinear problems are given.
Reviewer: E.D’jakonov

MSC:
65N15 Error bounds 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
35J25 Boundary value problems for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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