# Nédélec Elements and Newton-Methods

If you want to develop numerical algorithms for variational inequalities, you often choose a Semismooth Newton Method. In many cases, this approach involves derivatives of $\max$ or $\min$ functions in respective $L^p$-spaces, which result in pointwise evaluation of functions.

However, this isn't too much of a problem, if you're discretizing with, for example, Lagrange elements, since their degrees of freedom represent pointwise evaluation in the nodes.

Your algorithm and the Newton-steps obviously can't converge, if the DOFs are integrals of the tangential components over the edges! This is the case for Nédélec Elements for $H_0(curl, \Omega)$.

I wasn't able to find anything concerning this topic, so my question: Is there literature, that works around those problems or even an easy solution?

• Obviously, in this case you can't simply reduce the constraints on the function values to constraints on the coefficient vector and need to pay more attention to assembling the weak form of the pointwise Newton system. Some work has been done for control constraints, which have a similar structure, e.g., arxiv.org/pdf/1603.07099, dx.doi.org/10.1137/16M1074473 – Christian Clason Jan 30 '18 at 11:48
• – Christian Clason Jan 30 '18 at 12:04
• @ChristianClason Thank you for the references! – Malte Winckler Jan 31 '18 at 7:43

The constraints need to be based on reasonable physics. So if you are using a Nedelec element, then that's presumably because your physics are based on curls and tangential components, and any constraint you have is reasonably also only going to contain these. So the problem is not bigger than in other cases. If your problem requires pointwise continuity in the constraints, then you are either asking for something that's not physically reasonable, or you have chosen the wrong element.

• Actually, we are working with Maxwell variational inequalities with their physical origin in the superconducting-phenomena. By the nature of Maxwell's equations, the curl-conforming Nedelec-elements are the first choice. Could you please explain me, what you mean by "contraints" in this case? Thanks! – Malte Winckler Jan 31 '18 at 7:51
• My comment is about the origin of the inequality. The Maxwell equations are just an equality. If you are asking about semismooth methods, then you must have an additional effect you are trying to model that introduces the non-smoothness. My question is how this non-smoothness looks? How does it depend on the electric and magnetic field variables? – Wolfgang Bangerth Jan 31 '18 at 14:34
• Ah, okay. The non-linearity comes from Bean's Critical State model for superconductivity and gives us a variational inequality of the second kind (dependent on the electrical field)! – Malte Winckler Feb 1 '18 at 8:02
• I don't know the model. You may want to add the equations to your original question. – Wolfgang Bangerth Feb 2 '18 at 13:28

Forgive if I am wrong but in your question you confusing regularity of the solution as result of non-smooth coefficients in PDF, and approximation space which you choose appropriately to used operators in your week form.

As rule of thumb, if on your approximate field acts gradient operator you need H1 space, if divergence H-div, and H-curl curl operator. If to given filed no differential is applied, you can use L2 space.

It is the third component of the puzzle approximation base, which in principle can be independently selected from the space. Particular realisation of H-div and H-curl space with polynomial base are RT-elements and Nédélec elements. You can have approximation base with the singularity, where the solution is not regular at the point or on a curve. In plasticity or in the case mentioned by you (max function in coefficients), because of regularity of the solution you need low order elements.

If your coefficients in PDE have max term which depends on solution, then you can regularise problem, for example like this $$max(u^h,a) = (u^h + a + \frac{1}{r}| u^h-a |^r)/2$$ where $r\geq 1$, for example $r=1.01$. With that trick, your Newton method will work. You can look what is done in contact mechanics, where substantial work has been done to solve this type of problems, for example, Popp method of active set strategy.