Pressure boundary condition in lid driven cavity using finite element method

Question

Thank you all

1.) I am trying to solve the lid-driven cavity problem for an incompressible Stokes and Navier Stokes equations using the general "Mixed" finite element method. Dirichlet boundary conditions are specified everywhere on the boundary.

When I try to solve these systems of equations, my matrix showing is that it is singular. I doubt whether it is necessary to specify pressure value at some point. Otherwise how to impose the constraint "average pressure over the domain is zero"? in FEM.

2.) I am trying to simulate incompressible flow over a cylinder using the mixed finite element method. Dirichlet boundary conditions are specified at the inlet, top, and bottom on a circle and do nothing on the outlet.

Is it necessary to give pressure value at some point?

Answer 1

It makes sense that your matrix is singular, because the pressure is only known up to a constant. That is, if $u_h, p_h$ is a solution to your system of equations, then $u_h, p_h + C$ is also a solution.

To see that this implies singularity, note that if the saddle point linear system is written as $A(u,p) = b$, then $u_h, p_h = (0,0)$ is a solution to the system with $b=0$. But then $u_h, p_h = (0, 1)$ also solves the system with $b=0$, hence a non trivial nullspace and a singular linear system.

Without knowing the specifics of your scheme, it's hard to say exactly what you should do. But some general approaches you can take:

1. Add a row to your linear system enforcing that either the average of the mean element pressures is 0 over the domain, or that the pressure at some point in the domain is 0. Then solve the system in a least squares sense (using some iterative linear solver). This is probably the easiest approach.

2. Rather than solving the saddle point system, use an augmented lagrangian approach where pressure and velocity are decoupled and iterate between pressure and velocity guesses until convergence. If you start with a zero-mean guess for the pressure, the mean-zero property will be preserved. See this paper for a more detailed discussion. Decoupling pressure and velocity can have computational advantages as well, since it reduces the size of the linear systems you have to solve.

3. Rather than solve the saddle point system, use a projection method to decouple pressure and velocity (see Chorin, Guermond papers). In this case, you will have a singular pressure correction equation poisson equation, where you can apply a penalization method to remove the singularity.

Answer 2

Please check this post.

1. Zero mean pressure space is used for convenience when one is interested in FEA theory (basically, we cannot enforce $p(x_0) = p_0$ for $p \in \mathbb L^2$ since it does not make sense); from the computational point of view, it is easier to fix one of the pressure DOFs (although you can subtract mean value at the post–processing step if you want to). When you are working w/ polynomial spaces—and this is exactly what you do in FEM—it is perfectly fine to enforce $p(x_0) = p_0$. Handle this constraint like you usually handle Dirichlet BCs (e.g., via modifying your matrix).
   It is also fine to ignore this constraint in some cases (e.g., Krylov solvers can do fine with this).

2. In this case your matrix will be well-posed. Neumann condition fixes the issue with pressure defined up to a constant.

Answer 3

When you say using general "Mixed" finite element method, I assume that you are using the coupled velocity-pressure formulation. Correct me if I am wrong.

You have not specified whether you are using an LBB-stable element or a stabilised formulation. Nevertheless, irrespective of whether you use a stabilised formulation or LBB-stable element, when you apply Dirichlet BCs using elimination approach, it would help to fix the pressure at one point.

When you use the stabilised formulation, you may not face any convergence issues if you do not fix pressure at one node for some examples. But, when you use LBB-stable elements, you have to fix pressure at one node; you will have convergence issues if you don't.

If you are using the elimination approach to apply Dirichlet BCs, then you can simply eliminate the corresponding row and column of the fixed pressure node.

For the example problems you are interested in, I suggest to refer to this excellent PhD thesis on stabilised finite element formulations for incompressible fluid flow.

The text book by Donea and Huerta is also an excellent reference on mixed formulations for incompressible fluid flow.