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Some iterative methods of solving problems for parabolic equations. (English. Russian original) Zbl 0727.65076
Sov. Math., Dokl. 37, No. 3, 714-717 (1988); translation from Dokl. Akad. Nauk SSSR 300, No. 4, 782-785 (1988).
The iterative process $\frac{\partial u_{k+1}}{\partial t}-\alpha \Delta u_{k+1}=\frac{\partial}{\partial x_ i}[(A_{ij}(Pu_ k)- \alpha \delta_{ij})Q\frac{\partial u_ k}{\partial x_ i}]-A_ 0(Pu_ k,Qu_{k,k}),$ $$u_{k+1}(x,0)=\psi_ 0$$, $$u_{k+1}|_ S=\psi_ 1$$, is studied for the problem $$\partial u/\partial t-(\partial /\partial x_ i)(A_{ij}(u)\partial u/\partial x_ j)+A_ 0(u,u_ x)=0,$$ $$u(x,0)=\psi_ 0$$, $$u|_ S=\psi_ 1$$, where $$\alpha$$ is a parameter and P, Q are shear operators. Conditions for convergence are obtained. The author indicates that the iterative process can be applied to proving existence and evaluating some subset of solutions.
##### MSC:
 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35K15 Initial value problems for second-order parabolic equations
##### Keywords:
iterative methods; parabolic equations; convergence