PCG convergency
Hello,
I am developing FEM codes for 3D incompressible fluid flows. My current problem is that, for large scale problems (1M nodes), LU method is running extremely slow. It takes two days for one LU decomposition with FEM penalty method. An iterative solver (preconditioned conjugate gradient) was considered to be a solution. However, this case is difficult to converge especially in 3D if using this iterative solver. I am not sure if the problem is caused by the big number (inverse of penalty parameter) in the matrix. Any ideas and suggestions? Many Thanks! Steve 
Re: PCG convergency
I think usually the convergence is related with the problem itself, not the size. You can try a small problem to see if it converge or not. If not, try to load the matrix to MATLAB to check the eigenvalues. I use GMRES for usual 3D flow. It converges. But if I add free surface boundary condition, it doesn't converge.

Re: PCG convergency
The convergence speed of the PCG algorithm is directly related to the condition number of the matrix of the linear system.
In the penalty method, the greater epsilon (in div(v)=(1/epsilon)*p), the greater the condition number and the worse the convergence rate of the method. At that point there is nearly nothing else trivial to do, as far as I know, than trying to enhance the preconditioning of the matrix. Otherwise, you can switch to augmented lagrangian methods. Hope it could help, Anthony 
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