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

D. Z. Turner, K. B. Nakshatrala, K. D. Hjelmstad

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


In the following paper, we present a consistent Newton-Schur (NS) 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 NS 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.

Original languageEnglish (US)
Pages (from-to)119-137
Number of pages19
JournalInternational Journal for Numerical Methods in Fluids
Issue number2
StatePublished - Jan 2010
Externally publishedYes


  • Consistent Newton-Raphson
  • Incompressible Navier-Stokes
  • Schur complement
  • Stabilized finite elements
  • Three-dimensional flows
  • Variational multiscale formulation

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Computer Science Applications
  • Applied Mathematics


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