Jack,

Is it possible to extend your code to time dependent Navier-Stokes 
>> equations?
>>
>> I want to solve the unsteady Navier-Stokes equations using the 
>> *Vector-Valued* method, but I did not get any good idea from the 
>> tutorial examples (step-22, step-32, etc.) for Stokes flow.
>>
> The velocity in the Stokes and Navier-Stokes equations is vector-valued, 
but what exactly do you mean by "vector-valued method"?
step-22 considers the stationary Stokes equations and so does step-32. 
Basically, you just have to add a time discretization to your equations. 
Have a look at the examples that deal with time-dependent problems [1]. 
E.g. step-26[2] considers the heat equation using theta-time stepping 
scheme and might be a good starting point. Step-35[3] uses a projection 
method in combination with a BDF(2) time discretization to solve for the 
instationary Navier-Stokes equations.

Best,
Daniel

[1] https://www.dealii.org/8.4.1/doxygen/deal.II/Tutorial.html
[2] https://www.dealii.org/8.4.1/doxygen/deal.II/step_26.html
[3] https://www.dealii.org/8.4.1/doxygen/deal.II/step_35.html

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