Two Robust, Efficient, and optimally Accurate Algorithms for parameterized stochastic navier-stokes Flow Problems

TL;DR

This paper introduces two new finite element algorithms for parameterized Navier-Stokes flow problems, demonstrating their stability, convergence, and efficiency through theoretical analysis and numerical experiments, with a focus on uncertainty quantification.

Contribution

The paper develops and analyzes two fully discrete finite element algorithms for parameterized NSEs, incorporating ensemble methods, grad-div stabilization, and stochastic collocation for UQ, with proven stability and convergence.

Findings

Both algorithms are stable and convergent under certain conditions.

The splitting algorithm outperforms the coupled scheme in numerical tests.

The methods effectively integrate with stochastic collocation for uncertainty quantification.

Abstract

This paper presents and analyzes two robust, efficient, and optimally accurate fully discrete finite element algorithms for computing the parameterized Navier-Stokes Equations (NSEs) flow ensemble. The timestepping algorithms are linearized, use the backward-Euler method for approximating the temporal derivative, and Ensemble Eddy Viscosity (EEV) regularized. The first algorithm is a coupled ensemble scheme, and the second algorithm is decoupled using projection splitting with grad-div stabilization. We proved the stability and convergence theorems for both algorithms. We have shown that for sufficiently large grad-div stabilization parameters, the outcomes of the projection scheme converge to the outcomes of the coupled scheme. We then combine the Stochastic Collocation Methods (SCMs) with the proposed two Uncertainty Quantification (UQ) algorithms. A series of numerical experiments are given to verify the predicted convergence rates and performance of the schemes on benchmark problems, which shows the superiority of the splitting algorithm.