Associated people

J. Schröder, A. Schwarz, S. Serdas, in cooperation with S. Turek (TU Dortmund)

Abstract

The main focus of this project is to examine least-squares mixed finite elements for the solution of steady and unsteady Newtonian fluid flow, which is described by the incompressible Navier-Stokes equations.
Least-squares variational principles have increasingly gained attention, which is due to some theoretical and computational advantages compared to the Galerkin method.

• The method provides for instance an a posteriori error estimator without additional costs, which can be used for the development of adaptive mesh refinement algorithms.
• The resulting symmetric positive definite system matrices can be solved by using robust and fast iterative methods even for problems with governing nonselfadjoint operators such as fluid dynamics and transport problems.
• The inf-sup condition does not hold, so there are no restrictions for the choice of the polynomial degree of the finite element spaces.

The development and improvement of mixed least-squares finite element formulations by means of these topics are the main incentive of our research.
In general, we consider div-grad first-order systems resulting in approaches with e.g. stresses, velocities, and/or pressure as unknowns. The L₂-norms of the residuals of the derived equations yield then the least-squares functional, which is the basis for the associated minimization problem.

References

Cai, Z. and Lee, B. and Wang, P.[2004], “Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems”, SIAM Journal on Numerical Analysis 42, p. 843-859.

Schwarz, A. and Schröder, J. [submitted], “A mixed least-squares formulation of the Navier-Stokes equations for incompressible Newtonian fluid flow”, PAMM Proceedings in Applied Mathematics and Mechanics.
 
Münzenmaier, S. and Starke,G. [2011], “First-order system least squares for coupled Stokes-Darcy flow”, SIAM Journal on Numerical Analysis 49, p. 387--404.