A variational multiscale Newton-Schur approach for the incompressible Navier-Stokes equations

TL;DR

This paper introduces a Newton-Schur solution method for variational multiscale formulations of 3D incompressible Navier-Stokes equations, enhancing computational efficiency and scalability for complex, high-Reynolds-number flows.

Contribution

It develops a systematic, scalable Newton-Schur approach for variational multiscale Navier-Stokes problems, suitable for large, nonlinear, unstructured mesh computations.

Findings

Achieves quadratic convergence with Newton-Raphson scheme

Improves computational efficiency and parallel scalability

Successfully applied to 3D flows with Reynolds number up to 1000

Abstract

In the following paper, we present a consistent Newton-Schur solution approach for variational multiscale formulations of the time-dependent Navier-Stokes equations in three dimensions. The main contributions of this work are a systematic study of the variational multiscale method for three-dimensional problems, and an implementation of a consistent formulation suitable for large problems with high nonlinearity, unstructured meshes, and non-symmetric matrices. In addition to the quadratic convergence characteristics of a Newton-Raphson based scheme, the Newton-Schur approach increases computational efficiency and parallel scalability by implementing the tangent stiffness matrix in Schur complement form. As a result, more computations are performed at the element level. Using a variational multiscale framework, we construct a two-level approach to stabilizing the incompressible Navier-Stokes equations based on a coarse and fine-scale subproblem. We then derive the Schur complement form of the consistent tangent matrix. We demonstrate the performance of the method for a number of three-dimensional problems for Reynolds number up to 1000 including steady and time-dependent flows.