Weak Form for Navier-Stokes with Chorin's Projection

Описание: In order to solve the equations of fluid motion with FEniCS, we need to translate the Partial Differential Equations (PDEs) into their weak form together with a method to enforce incompressibility. Here are the notes: https://github.com/Ceyron/machine-lea...

The Navier-Stokes equations are the fundamental description for fluid mechanics. They are notoriously hard to solve numerically due to their saddle point structure by the incompressibility constraint. A Finite Element discretization by the FEniCS Python library requires a translation of the strong form into the weak form. In order to do so, we, however, first have to apply a projection method to enforce incompressibility given by the 2nd PDE. We will do this in strong form and then get a three-step algorithm for solving the Navier-Stokes equations. Finally, we have to translate all three strong PDEs into their weak form. This process involves the reverse product rule and an application of Gauss/Divergence theorem. All steps are presented in detail ;).

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Timestamps:
00:00 Intro
00:55 BC & IC for specific example
01:53 Agenda
02:13 Chorin's Projection overview (an operator splitting)
05:01 An algorithm in strong form
09:02 Obtaining an equation for pressure
13:23 Summary in strong form
15:41 (1) Weak form for tentative momentum step
30:11 (2) Weak form for Pressure Poisson problem
33:37 (3) Weak form for Velocity Projection/Correction
36:13 Summary in weak form
40:37 Outro