# Parallelizing FEM for elliptical PDEs with n >1

For a little personal project, I am picking up my FEM skills again. I learned a lot about the theory back in university and I am able to implement a simple FEM solver for specific problems but I was curious about the different parallelization possibilities because we never went into those too deeply.

Let us focus on the Poisson equation for simplicity in some "nice" domain for n >1. From what I have seen, one can

• Parallelize the assembly of the stiffness matrix. This takes quite some time for larger problems and can be parallelized easily.
• Parallelize the linear solver. Here there are lots of possibilities to construct specific solvers for sparse systems based on CG, multigrid or Krylov-based methods. Also possible but a little harder.
• Domain Decomposition. If I understood that topic correctly one can solve some sub-areas of the domain and divide the solution of that onto several processors.

I am not a researcher in the topic but would like to gain a deeper understanding in the topic — mainly not by using some available library (I am well aware of FEniCS et al.) but code the first simple examples for parallelization by hand. For example, I keep reading that DG FEM is supposed to scale a little bit better on a parallel level and from some slides I can guess why — you have smaller assumptions on continuity, so the faces of your elements do not need to be coupled strongly. I have however trouble finding some extensive discussion with relating examples about this.

I cannot find really well-explained material on the last topic either, I know it exists but I couldn't find comprehensive material on that topic either. Most jump from the formulation of the discretization to "… and then I used METIS plus xyz to solve. Here are three graphs showing convergence". It is of course also really tiring to try to reverse-engineer code found somewhere that "seems" to do what I think it does.

### Questions

• From those who know, are there known books on that topic I could read?

• Could someone point me in the right direction?

• Or would someone with some experience in those topics give a small introduction into the topic?