You may refer to:

Jean Donea, Antonio Huerta, Finite Element Methods for Flow Problems, Wiley.

Thank you Sumer Dirbude,

I have found a helpful part in Zienkiewicz Volume 1 page 560. He considers the same problem there, and provides that the discretization is a mixed formulation. I am wondering if it is necessary to treat it as such. As I am limited to using a FEM-Library that does not support mixed formulations/block solvers. Any idea how to tackle this?

I solved a poro-elasticity problem in the way you mentioned without using mixed FEM and it worked just fine. Several things you may want to keep in mind,

1. You need the convergence result before you do the iterative method. You also need to obtain a norm relationship between u and p such as \|u\| < C\|p\|

2. You ware trying to decouple the system here so make sure the regularity of u and p matches.

3. I'm assuming that you are solving a time dependent problem where each step involves a nonlinear steady equation. If so, you need to do a iterative method for each time step until it converges before you can move to the next time step.

I think you have already chosen the Mixed formulation, according to your attached manuscript, when you written the displacements and pressure degrees of freedom at each node.

Now the problem is relevant to the time integration algorithm you would  chose, probably for the kind of problem you would pick up an implicit algorithm (Newmark??). In that framework it is asked to solve the non-linearities  that can arise from the step n to the step n+1.

I'm not able to grasp the kind of time integration you used from your manuscript.

Dear Roberto,

thank you for your comments. I believe as well that this is given as a mixed formulation. But since I am limited in terms of what problems I can solve, solving the mixed system of equations simultaneously is not an option. I need to solve them iteratively. The time stepping will be of importance and I have found in Zienkiewicz - The finite element method- The Basis page 560 a guide on how to do correct time-stepping for this problem.

For me the main question is, can it be solved, by first solving for displacements and then for pressure in an iterative procedure, or do I have to solve them simultaneously, and if so, what is a good approach to do so?

Song Chen,

Thank you as well for your suggestions, they have led me to useful considerations.

TO verify that I get it correct I say:

- If a solution  exists to the coupled system it is possible to find it out iteratively.

- Some conditions shall be satisfied by the matrix K and Q to hold the previous statement. Anyway the iterations, we are talking about, shall be performed "intra" the time integration  step

- For me the main question is, can it be solved, by first solving for displacements and then for pressure in an iterative procedure, or do I have to solve them simultaneously, and if so, what is a good approach to do so?

Maybe I was not clear about this part. What I mean is that this is doable as long as you have the convergence result for sequence u_n, p_n. For example, if I understand it correctly, you are solving by constructing two sequences u_n, p_n by solving p_n from 

Ku_{n-1}+Qp_n = f(u_{n-1})

then for u_n from some equation

A(u_n) = g(p_n)

and use this one again to find p_{n+1} until the sequences converge. This will work as long as both sequences u_n and p_n converge to the exact solution u,p, which can be proved by finite element analysis. Please check page 6-8, Theorem 2.2 in the following file. The key step for doing so is equation (2.14). Meanwhile, if you are setting the stopping condition for the iterative method to be \|u_n-u_{n-1}\|

Hello Lukas Mosser,  I have not been able to access the link you attached hence I was unable to view Eqn. 17. Could you please provide your system of PDEs as a word or pdf attachment? Nevertheless, I believe a mixed FEM is the best procedure to adopt. When implemented in an implicit scheme it is unconditionally stable. I will provide the FE discretisation equations once I know the governing PDEs.

Hi Lukas,

you are not obliged to solve your linear system as a block; it may indeed be a good idea to use your iterative process; be aware however that your iterations may not converge, this depends on the iteration matrix that relates your error at iteration k to the error at iteration k-1 (the error being here the difference between your solution at iteration k and the exact solution of the system).

If your process does not converge, you may want to use under relaxation, i.e. say that your iteration at step k+1 is a convex combination of the iteration at step k and the result of your linear solves.

Hope this helps,

Pascal

You can use the quasi linearisation technique associated with other approch as finite difference method or FEM.

Dear all,

Thank you very much for your comments they have been very helpful and have led me into a number of interesting and possibly fruitful directions.

Since this is an ongoing project, I will let you know what I used as a solution when I have managed to implement everything.

Best regards,

Lukas