ABSTRACT
This chapter presents asymptotically optimal algorithms for solving boundary value problems associated with systems of Stokes or Navier-Stokes type. Such equations are of fundamental importance in the theory of elasticity and shells, hydrodynamics, meteorology, magnetohydrodynamics, and other fields of science. The chapter describes assumptions on the parameters that are typical for mathematical analysis of the correctness of such problems. More cumbersome is the proof of the second assertion, but those difficulties are analogous to those for elliptic problems with bounded power nonlinearity. Some generalizations are quite transparent for certain nonlinear problems of thermohydrodynamics arising in meteorology, when the incompressibility condition is preserved. Numerical methods, especially mixed finite element methods, for problems of Stokes or Navier-Stokes types have been studied by many. But questions related to the construction of asymptotically optimal algorithms are especially difficult and significant progress in this direction is fairly and for results with multigrid methods.
