bugfix: rwg projectors

[PaulOlyslager]

• switched DualLoops and DualStars
• fixed iterative projectors for rwg

[krcools]

@cmuenger are you happy with this?

[cmuenger]

I'm a bit concerned about the inconsistency between using the projectors for RT and GWP basis functions. With @PaulOlyslager's changes in this PR, the iterative computation of the projectors for RT basis functions now yields the actual projectors, which is probably what a user expects.

However, for the computation to be efficient for higher-order basis functions, the computation is off by a factor of a Gram matrix (to avoid the computation of the inverse square root of a matrix). Thus, the higher-order projectors must be compensated by an inverse Gram matrix placed between them. This has to be done actively when using higher-order projectors. This does not affect the Bourhis Theta, since the Theta operator contains the Gram matrices in between the projectors.

In the lowest-order case, the Gram matrix whose square root is needed appears to be diagonal, see Paul's changes.

Regarding the name change, do we now have the following naming convention? @PaulOlyslager
Stars: Local stars -> P\Sigma = \Sigma (\Sigma^T*\Sigma)^+ \Sigma^T
Loops: Local loops + global loops -> P\Lambda H = I - P \Sigma

Dual stars: Local loops -> \mathbb{P}\Lambda = \Lambda(\Lambda^T\Lambda)^+\Lambda^T
Dual loops: Local stars + global loops -> \mathbb{P}\Sigma H = I-\mathbb{P}\Lambda

For clarity, the convenience functions P\Lambda and \mathbb{P}\Sigma should be renamed to P\LambdaH and \mathbb{P}\SigmaH. This issue goes back to my initial code, where I did not make this proper distinction. But if we improve the usability of the qH-projector code in this PR, we should also include this small tweak.

[PaulOlyslager]

I think this inconsistency has its roots in the theory, I just made sure that if you use iterative or direct on the projection object you end up with the same operator. Maybe we can introduce an abstract type: l2 projector and L2 projector because that is in essence the difference.

The naming convention is indeed, I agree to add the letter H for clarity.
Stars: Local stars -> P\Sigma = \Sigma (\Sigma^T*\Sigma)^+ \Sigma^T
Loops: Local loops + global loops -> P\Lambda H = I - P \Sigma

Dual stars: Local loops (on primal mesh) -> \mathbb{P}\Lambda = \Lambda(\Lambda^T\Lambda)^+\Lambda^T
Dual loops: Local stars (on primal mesh) + global loops -> \mathbb{P}\Sigma H = I-\mathbb{P}\Lambda

[cmuenger]

Yes, consistency between direct and iterative is more important than between different basis functions.

I think the abstract types are a good idea. It allows for flexibility to implement l2 or L2 projectors in other cases. We probably should document/ add comments to the code to clarify this difference between the projectors for RWG and GWP.

If we introduce the l2 and L2 projectors, should the initial implementation I did be revived? However, not as the default as Paul's implementation follows the "standard" framework for quasi-Helmholtz projectors.