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The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposi- tion methods and reduced basis discretizations. The basic idea is to first decompose the computational domain into a series of subdomains that are deformations of a few reference domains (or generic com- putational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation, but solved for different choices of deformations of the reference subdomains and mapped onto the reference shape. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the generic computational part to the actual computational part. 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.

A reduced basis element method for the steady Stokes problem