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A reduced basis element method for the steady Stokes problem: Application to hierarchical flow systems

机译：稳定Stokes问题的简化基元方法：应用于分层流系统

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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 decomposition methods and reduced basis discretizations (Fink & Rheinboldt (1983), Noor & Peters (1980), Prud'homme et at. (2002)), and its applications extend to, for example, control and optimization problems. The basic idea is to first decompose the computational domain into a series of subdomains that are similar to a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation, but solved for different choices of some underlying parameter. In this work, the parameters are representing the geometric shape associated with a computational part. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the reference domain for the part to the actual domain. We extend earlier work (Maday & Ronquist (2002), Maday & Ronquist (2004)) in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to constructing the basis functions, to the mapping of the velocity fields, to satisfying the inf-sup condition, and to "gluing" the local solutions together in the multidomain case (Belgacem et al. (2000)). We also demonstrate an algorithm for choosing the most efficient precomputed solutions. Two-dimensional examples are presented for pipes, bifurcations, and couplings of pipes and bifurcations in order to simulate hierarchical flow systems.
机译：简化的基元方法是一种新的方法，用于近似求解偏微分方程描述的问题。该方法源于域分解方法和简化的基础离散化（Fink＆Rheinboldt（1983），Noor＆Peters（1980），Prud'homme等人（2002）），其应用扩展到例如控制和控制。优化问题。基本思想是首先将计算域分解为与几个参考域（或通用计算部分）相似的一系列子域。与每个参考域相关联的是对应于相同的主导偏微分方程的预计算解，但是针对某些基础参数的不同选择进行了求解。在这项工作中，参数表示与计算部分关联的几何形状。然后，将对应于新形状的近似值视为预先计算的解的线性组合，从该零件的参考域映射到实际域。我们在此方向上扩展了早期工作（Maday＆Ronquist（2002），Maday＆Ronquist（2004）），以解决由稳定斯托克斯方程控制的不可压缩流体流动问题。在多域情况下，特别着重于构造基函数，速度场的映射，满足扰动条件以及将局部解“粘合”在一起（Belgacem等人（2000））。我们还演示了一种用于选择最有效的预计算解决方案的算法。给出了管道，分叉以及管道和分叉的耦合的二维示例，以模拟分层流系统。

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