zbMATH — the first resource for mathematics

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.
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