Solving the stokes problem on a massively parallel computer

Abstract

We describe a numerical procedure for solving the stationary two‐dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distributed memory architecture. Numerical experiments on a Cray T3E computer illustrate the parallel performance of the method. Stokso uždavinio sprendimas galingais lygiagrečiaisiais kompiuteriais Santrauka Aprašomas skaitinis metodas stacionariajam dvimačiam Stokso uždaviniui. Metodas pagristas baigtiniu elementu aproksimacija greičiui ir slegiui, stabilumo reguliarizacija ir defektu taisymo metodu, kuris pagerina tiksluma. Eliminuojant nežinomus greičius iš algebrines lygčiu sistemos slegiui surasti gaunama simetrine teigiamai pusapibrežtine sistema, kuri sprendžiama vidinemis‐išorinemis iteracijomis. Išorine iteracija naudoja besalygini jungtiniu gradientu metoda. Vidines iteracijos, kuriu kiekviena atitinka kraštinio elipsinio uždavinio sprendima kiekvienai greičio komponentei, naudoja salygini jungtiniu gradientu metoda. Žinant slegi surandamas greitis. Metodas yra ekonomiškas, nes kompiuterio skaičiavimai proporcingi nežinomuju skaičiui. Metodas pritaikytas išnaudoti galingu lygiagrečiuju kompiuteriu su paskirstyta atmintimi architektūra. Skaitiniai eksperimentai kompiuteriu Cray T3E iliustruoja metodo išlygiagretinima. First Published Online: 14 Oct 2010

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