On rapidly converging iterative methods with incomplete splitting of boundary conditions for a multidimensional singularly perturbed system of Stokes type

B. V. Pal'tsev

Sbornik. Mathematics, Tome 81 (1995) no. 2, pp. 487-531

Résumé

Constructed and investigated are iterative methods for solving the Dirichlet problem for a system with small parameter $\varepsilon >0$: $$ -\varepsilon^2\Delta\mathbf{u}+\mathbf{u}+\operatorname{grad}p=\mathbf{f},\qquad \operatorname{div}\mathbf{u}=0, $$ leading at each iteration to splitting into a Neumann problem for the pressure and a vector Dirichlet–Neumann problem for the velocities. The case of periodic 'flows' between parallel walls is studied. The fastest variants of the method have the rate of convergence of a geometric progression with ratio of order $\varepsilon$. Also obtained are '$\varepsilon$-coercive' estimates of the solutions of the original problem in Sobolev norms.

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