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Methods of finite differences and finite elements of extra orders of accuracy are developed.
Some modifications of incomplete decomposition of matrixes on triangular multipliers for the solution of systems of linear algebraic equations received as a result of digitization of equations of mathematical physics are created. The idea of cyclic reduction has increased opportunities of paralleling and has lowered the level of recursive accumulation of mistakes comparing to consecutive (incomplete) elimination of unknown. A new method of dipole compensation (indemnity) is offered to reduce the consequences of suppression of small items. It is more exact for some problems than the method of diagonal compensation (indemnity) and many times more exact than a standard method of suppression of small extradiagonal elements.
For the equations of elliptic type with interface layers and parabolic equations with implicit time approximation a high degree of paralleling of the created algorithms and their higher accuracy comparing to other similar algorithms is proved and checked up experimentally on computing cluster. The number of arithmetic operations in these algorithms is proportionate to the number of unknown.
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