[Dune] Handling of mortar crosspoints

[Max Sagebaum]

Hello Andreas,

I could circumvent the problem by "removing" the Mortar degrees of freedom from the assembled matrix (Eq. 1.46 and 1.49 from the Book below). I tested it with a case where I have one block and a second case where the same block is split into 4 subblocks, which introduce crosspoints. With both Systems I get the same solutions and convergence rate of the linear system solve. Therefore, I do not have a direkt need for the handling of crosspoints anymore.

Nevertheless, what one would need, would be an implementation of "B. Wohlmuth: Discretization Methods and Iterative Solvers Based on Domain Decomposition"  (https://link.springer.com/book/10.1007%2F978-3-642-56767-4)<https://link.springer.com/book/10.1007%2F978-3-642-56767-4> Equation 1.31. For triangles there exist direct formulations how these modifications can be applied. For the quadliteral case it is more involved since the dual Lagrange functions are defined by the finite element and not on the reference element. Maybe a simple solution can be found when the crosspoint handling is implemented for the dual langrange basis  on the reference element like in Figure 1.15. This could be done by defining for the basis which nodes are crosspoints and which are not. If this is used with a special  transformation of the dual lagrange function like in "Stable Lagrange multipliers for quadrilateral meshes of curved interfaces in 3D" Equation 21 (https://www.sciencedirect.com/science/article/pii/S0045782506002866?via%3Dihub), then a proper handling of the crosspoints might be possible. I can not say for sure, since FEM methods are not my field of expertise and I currently have not time to dig further into the theory.

Cheers

Max

On Mon, 2021-07-05 at 09:30 +0000, Dedner, Andreas wrote:
Hi Max.
I'm not aware of anyone having used mortar methods with dune-fem. What do you need to get it to work?
Best
Andreas

________________________________
From: Dune <dune-bounces at lists.dune-project.org> on behalf of Max Sagebaum <max.sagebaum at scicomp.uni-kl.de>
Sent: 30 June 2021 13:35
To: dune at lists.dune-project.org <dune at lists.dune-project.org>
Subject: [Dune] Handling of mortar crosspoints

Hello @ all,

I am using  the DUNE FEM assembly to create the FEM Matrix for a simple Laplace equation. The mesh consists of several blocks, that are connected via 1to1 interface or unstructured interfaces (Abutting1to1 or Abutting). I use the mortar method in order to handle the connections between these blocks.

I am now facing the challenge, that I have to handle the crosspoints of the mortar interface (the points/edges where multiple mortar interfaces meet).  I know the theory that the Dual Langrange elements have to be modified for elements that contain such crosspoints. Before I start to implement these modifications myself, I wanted to ask if this is already available in DUNE?

Thanks

Max Sagebaum

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