On iterative methods for some elliptic equations with nonlocal conditions

Abstract

Abstract. The iterative methods for the solution of the system of the difference equations derived from the elliptic equation with nonlocal conditions are considered. The case of the matrix of the difference equations system being the M-matrix is investigated. Main results for the convergence of the iterative methods are obtained considering the structure of the spectrum of the difference operators with nonlocal conditions. Furthermore, the case when the matrix of the system of difference equations has only positive eigenvalues was investigated. The survey of results on convergence of iterative methods for difference problem with nonlocal condition is also presented.

Application of M-matrices theory to numerical investigation of a nonlinear elliptic equation with an integral condition

“…Let us write the eigenvalue problem for matrix A AU = λU https://www.mii.vu.lt/NA in the form closer to system (4)-(6) (see, e.g. [26,33])…”

“…It has been proved [33] that all the eigenvalues of (22), in the case γ > 0, are positive if and only if 0 γ < γ 0 ≈ 3.42. The exact value of γ 0 is…”

“…In many cases, the matrix of system of difference equations approximating an elliptic equation with nonlocal conditions has the properties typical to M-matrices. Convergence of iterative method to some systems of linear difference equations using the M-matrix properties was analysed in [33].…”

Semi-Implicit Difference Scheme for a Two-Dimensional Parabolic Equation With an Integral Boundary Condition

“…Application of the M -matrices in the convergence of iterative methods of difference problems with nonlocal conditions, arising from elliptic equations have been considered in [13,36].…”

A survey on stationary problems, Green's functions and spectrum of Sturm–Liouville problem with nonlocal boundary conditions

“…Štikonienė, Sapagovas andČiupaila consider convergence of iterative processes for elliptic differential problem with nonlocal conditions [219].…”