Solvability and convergence of nonlinear difference schemes. (English. Russian original) Zbl 0678.65038

Ardelyan, N. V.

Sov. Math., Dokl. 38, No. 2, 405-408 (1989); translation from Dokl. Akad. Nauk SSSR 302, No. 6, 1289-1292 (1988).

A nonlinear operator \(P_ h\) is given, mapping one Banach space into another and depending on a parameter h. The principal theorem is stated without proof, and it may be summarized by saying that if the Fréchet derivative of \(P_ h\) has a uniformly bounded inverse on a set \(\Omega\), then the equation \(P_ hu=0\) has a unique solution on \(\Omega\). The theorem is applied to finite-difference equations for a quasilinear elliptic partial differential equation, and in this setting h is the mesh size.

Reviewer: G.Hedstrom

MSC:

65J15	Numerical solutions to equations with nonlinear operators
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
47J25	Iterative procedures involving nonlinear operators
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

convergence; nonlinear difference schemes; Banach space; finite- difference equations; quasilinear elliptic partial differential equation