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Abstract
This paper deals with the stability analysis of multicomponent iterative methods for solving elliptic problems. They are based on a general splitting method, which decomposes a multidimensional parabolic problem into a system of one dimensional implicit problems. Error estimates in the L 2 norm are proved for the first method. For the stability analysis of Seidel type iterative method we use a spectral method. Two dimensional and three dimensional problems are investigated. Finally, we present results of numerical experiments. Our goal is to investigate the dependence of convergence rates of multicomponent iterative methods on the smoothness of the solution. Hence we solve a discrete problem, which approximates the 3D Poisson's problem. It is proved that the number of iterations depends weakly on the number of grid points if the exact solution and the initial approximation are smooth functions, both. The same problem is also solved by the Stability Correction iterative method. The obtained results indicate a similar behavior.
Šiame straipsnyje tesiama daugiakomponentiniu iteraciniu metodu stabilumo analize, pradeta ankstesniuose autoriu darbuose. Irodytas vienos schemos sprendinio konvergavimas L 2 normoje. Spektriniu metodu ištirtas dvieju Zeidelio tipo iteraciniu schemu stabilumas dvimačiu atveju, irodyta, kad dvimačiu atveju abi schemos yra nesalygiškai stabilios. Trimačio uždavinio spektrinio stabilumo analize atlikta skaitiškai. Irodyta, kad modifikuotoji schema pasižymi didesniu konvergavimo greičiu.
Paskutiniame skyriuje pateikti skaičiavimo eksperimento rezultatai. Buvo sprendžiamas trimatis Puasono uždavinys, aproksimuotas standartine baigtiniu skirtumu schema. Ištirta iteraciniu metodu konvergavimo greičio priklausomybe nuo sprendinio ir pradinio artinio glodumo. Parodyta, kad baigtiniu skirtumu schemoms konvergavimo greitis gali silpnai priklausyti nuo diskrečiojo tinklo mazgu skaičiaus, jei pradine paklaida yra glodi funkcija. Daugiakomponentiniai iteraciniai metodai palyginti su stabilizuojančios pataisos metodu, kuris irgi yra nesalygiškai stabilus.