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V.V. Akimov

Additive Schwarz preconditioner for the Neumann problem with a boundary layer

In this paper we consider a variant of the additive Schwarz preconditioner for elliptic finite element problems on meshes which are strongly refined in the vicinity of the Neumann boundary condition. Local refinements can be motivated by a sharp boundary layer in the solution of the underlying differential equation or they may occur in a method for solving another differential problem. The classical example of the latter situation is the L ₂-projection (or the splitting-up) method for the Navier–Stokes equations.

For a model problem we introduce an additive Schwarz preconditioner and prove that the condition number of the preconditioned stiffness matrix is bounded from above by a constant which is independent of the mesh and the width of the boundary layer. We consider several generalizations and applications of the results to the model problem. In particular, we discuss the application of the proposed preconditioner in the mortar element method. Numerical computations confirm the theoretical results of the paper.

Russian Journal of Numerical Analysis and Mathematical Modelling, Walter de Gruyter

Print ISSN: 0927-6467
Volume: 18, 06/2003
Pages: 185 - 206

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