Consistent Newton-Raphson vs. fixed-point for variational multiscale formulations for incompressible Navier-Stokes

TL;DR

This paper compares the efficiency and convergence of consistent Newton-Raphson and fixed-point iteration methods for solving variational multiscale formulations of incompressible Navier-Stokes equations, demonstrating superior performance of the Newton-Raphson approach.

Contribution

It introduces a consistent linearization of Navier-Stokes equations within a variational multiscale framework and compares two solution strategies, highlighting advantages of the Newton-Raphson method.

Findings

Newton-Raphson converges in fewer iterations.

Consistent formulation converges where fixed-point diverges.

Effective for Reynolds numbers up to 5000.

Abstract

The following paper compares a consistent Newton-Raphson and fixed-point iteration based solution strategy for a variational multiscale finite element formulation for incompressible Navier-Stokes. The main contributions of this work include a consistent linearization of the Navier-Stokes equations, which provides an avenue for advanced algorithms that require origins in a consistent method. We also present a comparison between formulations that differ only in their linearization, but maintain all other equivalences. Using the variational multiscale concept, we construct a stabilized formulation (that may be considered an extension of the MINI element to nonlinear Navier-Stokes). We then linearize the problem using fixed-point iteration and by deriving a consistent tangent matrix for the update equation to obtain the solution via Newton-Raphson iterations. We show that the consistent formulation converges in fewer iterations, as expected, for several test problems. We also show that the consistent formulation converges for problems for which fixed-point iteration diverges. We present the results of both methods for problems of Reynold's number up to 5000.