I am interested in reading about the analysis (existence, uniqueness, error estimates) of elliptic problems solved with a Mixed method that uses the Raviart Thomas elements (so far so good, easy to find) but with the presence of hanging nodes, (I haven't found anything). I have a paper from 2002 that claims this that this was a then-current open problem. Even just a proof of the inf-sup condition would be appreciated.

Is this still open? I would really appreciate a link to a paper.

(The normal Raviart Thomas projection doesn't work (as far as I can tell) in the proof of the inf-sup condition.



1 Answer 1


I think it is an interesting question. I do not have a specific paper to cite for you. But I am familar with the usage of hanging nodes with Nedelec elements for electromagnetism, and there is some care required regarding the continuity constraints for elements in the vicinity of a hanging node. For mundane H-grad functions, you have to enforce continuity of the solution. For H-curl (Nedelec) it is similar, but you should enforce continuity only for the tangential component (the normal components can/must "float" freely across the hang). When both spaces are in play (for instance, using H-grad for scalar potential and H-curl for electric field) you must enforce both these continuities, which will guarantee that your E-field functions can span the gradients of your potentials (this is essentially the motivation behind the Nedelec functions to begin with, to model the nullspace of the curl operator correctly, the "exact sequence" concept).

I would expect similar "compatibility conditions" to be your guide when using Raviart-Thomas / H-div functions. I'd think that upon the elements near the hanging node you should enforce normal continuity and let the tangential component "float". I have successfully used this approach to design stable subgridding schemes for FDTD (E-field uses H-curl functions with tangential continuity enforced at the hang, and B-field uses H-div functions with normal continuity enforced at the hang, setting up a similar compatibility between curl of E and span of B). I can point you to my thesis for some details .. but all this CEM might be wandering a bit afield from the PDE that motivated your original question (which sounds, hmm, Stokesy? not really my area).

Title: H-, P- and T-Refinement Strategies for the Finite-Difference-Time-Domain (FDTD) Method Developed via Finite-Element (FE) Principles

Linky: https://etd.ohiolink.edu/pg_10?0::NO:10:P10_ACCESSION_NUM:osu1219064270


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