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

The iterative methods to solve the system of the difference equations derived from the nonlinear elliptic equation with integral condition are considered. The convergence of these methods is proved using the properties of M-matrices, in particular, the regular splitting of an M-matrix. To our knowledge, the theory of M-matrices has not ever been applied to convergence of iterative methods for system of nonlinear difference equations. The main results for the convergence of the iterative methods are obtained by considering the structure of the spectrum of the two-dimensional difference operators with integral condition. *The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014)

Convergence of Iterative Methods for Nonlinear Elliptic Equation with Nonlocal Condition and M-Matrices

Šiame darbe nagrinėjami iteraciniai metodai skirtuminėms lygtims, gautoms iš dvimačių netiesinių elipsinių lygčių su nelokaliąja integraline ir Dirichlė kraštinėmis sąlygomis. Įvade apžvelgiami ankstesni šia tematika parašyti straipsniai ir knygos. Antrajame skyriuje suformuluojamas kraštinis uždavinys netiesinei elipsinei diferencialinei lygčiai su nelokaliąja integraline sąlyga, užrašoma atitinkama skirtuminė schema bei šios schemos matricinis pavidalas. Trečiajame skyriuje aprašomi darbe naudojami apibrėžimai ir žymėjimai bei apžvelgiamos pagrindinės M-matricų savybės. Pagrindinis darbo rezultatas nagrinėjamas 4 skyriuje, kuriame, priėmus tam tikras sąlygas suformuluojamos ir įrodomos trys teoremos apie iteracinių metodų konvergavimą. Toliau nagrinėjamas tikrinių reikšmių uždavinys ir iteracinių metodų konvergavimo srities praplėtimo klausimai. Paskutiniame skyriuje aprašomi trys konkretūs iteraciniai metodai ir atliekami skaičiavimai vienmačiu uždavinio atveju.This paper analyzes iterative methods for differential equations obtained from two-dimensional nonlinear elliptic equations with nonlocal integral and Dirichlet boundary conditions. The introduction provides an overview of the articles and books written on this topic. In the second section the boundary problem for nonlinear elliptic difference equation with nonlocal integral condition is formulated and the relevant difference scheme and the matrix form of this scheme is recorded. The third section describes the definitions and notations used in this paper and presents an overview of the main properties of M-matrices. The main result of the work is analyzed in section 4, where, based on certain conditions, three theorems about the convergence of iterative methods are formulated and proved. Next, an eigenvalue problem and the issues concerned with the expansion of the convergence range of iterative methods are analyzed. In the last section three specific iterative methods are described and the calculations of one-dimensional problem are performed.Informatikos fakultetasVytauto Didžiojo universiteta

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

The iterative methods to solve the system of the difference equations derived from the nonlinear elliptic equation with integral condition are considered. The convergence of these methods is proved using the properties of M-matrices, in particular, the regular splitting of an M-matrix. To our knowledge, the theory of M-matrices has not ever been applied to convergence of iterative methods for system of nonlinear difference equations. The main results for the convergence of the iterative methods are obtained by considering the structure of the spectrum of the twodimensional difference operators with integral conditionMatematikos ir statistikos katedraVilniaus universitetasVytauto Didžiojo universiteta