Collocation Methods for the Navier-Stokes Equations

This book deals with the numerical approximation of the time-dependent Stokes and the incompressible Navier-Stokes equations in primitive variables with periodic boundary conditions. It covers theory for the equations themselves as well as a thorough numerical analysis for the novel collocation methods which are introduced in this work.

After a projection step, the analytical equations are transformed into an abstract parabolic problem on which the further investigation is focused. A method of lines solution is discussed which uses collocation in space to separate the space and time variables. It is based upon deliberately chosen matrix-valued kernel functions which allow an analytically divergence-free approximation of the velocity field. Precise error and stability results for the velocity field and the pressure function are derived, and further time discretizations as well as details on a fast implementation are provided. Numerical experiments are performed in order to verify the theoretical claims from the convergence analysis and also to gain insights into the overall performance of the schemes.