A reduced basis element method for the steady Stokes problem

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Abstract: The reduced basis element method is a new approach for approxi- mating the solution of problems described by partial differential equa- tions. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decom- pose the computational domain into a series of subdomains that are deformations of a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solu- tions corresponding to the same governing partial differential equation, but solved for different choices of deformations of the reference sub- domains and mapped onto the reference shape. The approximation corresponding to a new shape is then taken to be a linear combina- tion of the precomputed solutions, mapped from the reference domain for the part to the actual domain. We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to satisfying the inf-sup condition, to a posteriori error estimation, and to "gluing" the local solutions together in the multidomain case.

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Date: 2004-12-02