โ Scribed by S.V. Sivashinskii
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Projection difference schemes for sy~~etric positive first-order systems, based on multilinear mesh functions, are stUdied. It is shown that the schemos have accuracy of order h'l,
The efficiency of a two-step iterative method for solving the difference equations is proved.
The present paper we consider a finite difference scheme, constructed by Galerkin's method using multilinear coordinate mesh functions. The direct use of Galerkin's method, providing a simple regular way of obtaining the difference equations and a scheme of low connectivity, leads in general to poor conditionality of the system of difference equations. Hence the integral of lowest terms is replaced in Galerkin's method by a sum, and a norm is introduced in a special way into the space of mesh functions. This device was used earlier in /1/ in projection schemes to solve elliptic equations. The modification of Galerkin's method does not result in a reduction of the order of accuracy of the scheme. We show that, in spite of using multilinear mesh functions, the scheme has order of accuracy 11" , where h is the mesh step, due to its symmetry.
A two-step iterative method is used to solve the difference equations. Such methods were considered in /2/ and /3, Chapter 8, Sect.3/. A specific feature of the version employed is the asymmetry of the resolving operator of the transition of the internal process. OUr main result is Theorem 2, in which we establish that of the order of h-'(ln 11-')' iterations are required to obtain the approximate solution with accuracy of order h~.
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