an:03850423
Zbl 0535.65071
Hackbusch, Wolfgang
On the regularity of difference schemes. II: Regularity estimates for linear and nonlinear problems
EN
Ark. Mat. 21, 3-28 (1983).
00148237
1983
j
65N15 65N12 35J05 35J25 35J65
regularity estimates; a priori estimates; coercitivity estimates; Poisson equation
[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 H??lder 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.
E.D'jakonov
Zbl 0462.65058