While approximating the domains in order to apply the finite element methods for Navier Stokes Equations, we have used 2 different meshes for the same domain.
In general why we may need 2 meshes for the same domain (our 2 variables are $u$ the velocity of the fluid and $p$ the pressure of the fluid)? Is it to ensure stability? If yes, how could I know what mesh I have to choose to ensure stability?
Well if i remember correctly we use two meshes for the same domain to make sure that velocity-pressure coupling satisfy the LBB condition. Like we use a Q1Q1 or Q2Q1 bubble function elements to solve such problems since these elements satisfy the LBB condition. Since it's a very vast topic I have only mentioned in the general scene. If you want to look for detailed explanation i would recommend you to read the chapter 6 of the book "Finite Element Methods for Flow Problems" by Jean Donea and Antonio Huerta.
I have the original copy of this book so if you could't find the soft copy let me know i can send you a pdf copy of the book.