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

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

Research output: Contribution to journalArticle

4 Citations (Scopus)

### Abstract

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 language English (US) 119-137 19 International Journal for Numerical Methods in Fluids 62 2 https://doi.org/10.1002/fld.2014 Published - Jan 2010 Yes

### Fingerprint

Incompressible Navier-Stokes Equations
Navier Stokes equations
Schur Complement
Tangent line
Stiffness matrix
Computational efficiency
Variational multiscale Method
Nonsymmetric Matrix
Scalability
Three-dimensional
Reynolds number
Newton-Raphson
Formulation
Unstructured Mesh
Stiffness Matrix
Computational Efficiency
Three-dimension
Navier-Stokes Equations
Nonlinearity

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

A variational multiscale Newton-Schur approach for the incompressible Navier-Stokes equations. / Turner, D. Z.; Nakshatrala, K. B.; Hjelmstad, Keith.

In: International Journal for Numerical Methods in Fluids, Vol. 62, No. 2, 01.2010, p. 119-137.

Research output: Contribution to journalArticle

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