Abstract
Incompressible fluid dynamics is a challenging topic, basically for the “saddle point” nature of the problem. Under certain assumptions, the velocity field that solves this problem is the minimum of an energy constrained by incompressibility. This makes this problem significantly different from elliptic problems (corresponding to free minimization). Pressure is the Lagrange multiplier associated with the incompressibility constraint. For this reason, the numerical solution may be in general expensive to compute and large efforts have been devoted to develop efficient solution schemes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
It is sufficient that f be a member of V′.
- 2.
Freefem++ denotes the space coordinates with x and y.
- 3.
In general, for a system like (7.18), matrix —DK -1DT is said pressure Schur complement of the system.
- 4.
We solve this system with a direct method. For ∈ = 0 Freefem++ clearly would give an error message, since the matrix is singular.
- 5.
Different possible implementations of this preconditioner are available. In Freefem++ manual a different implementation is presented.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Italia
About this chapter
Cite this chapter
Formaggia, L., Saleri, F., Veneziani, A. (2012). Navier-Stokes equations for incompressible fluids. In: Solving Numerical PDEs: Problems, Applications, Exercises. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2412-0_7
Download citation
DOI: https://doi.org/10.1007/978-88-470-2412-0_7
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-2411-3
Online ISBN: 978-88-470-2412-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)