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arXiv · 1710.05812

A Low-rank solver for the Navier--Stokes equations with uncertain viscosity

Kookjin Lee·Howard C. Elman·Bedřich Sousedík
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Abstract

We study an iterative low-rank approximation method for the solution of the steady-state stochastic Navier--Stokes equations with uncertain viscosity. The method is based on linearization schemes using Picard and Newton iterations and stochastic finite element discretizations of the linearized problems. For computing the low-rank approximate solution, we adapt the nonlinear iterations to an inexact and low-rank variant, where the solution of the linear system at each nonlinear step is approximated by a quantity of low rank. This is achieved by using a tensor variant of the GMRES method as a solver for the linear systems. We explore the inexact low-rank nonlinear iteration with a set of benchmark problems, using a model of flow over an obstacle, under various configurations characterizing the statistical features of the uncertain viscosity, and we demonstrate its effectiveness by extensive numerical experiments.

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Kookjin Lee, Howard C. Elman, Bedřich Sousedík. 2017-10-16. A Low-rank solver for the Navier--Stokes equations with uncertain viscosity. https://doi.org/10.1137/17m1151912

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