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I've got a uniform quadrangular mesh and for each node there's a vector quantity $u$ defined. I also have a non-aligned material interface across the mesh. Now I need that vector quantity to have a zero jump across the interface $[u\cdot n]=0$ where $n$ is the interface normal. I want to project $u$ into another space that satisfies this condition. Could I use Raviart-Thomas finite element for this case? I have never worked with RT elements and I was wondering if they could be built using an arbitrary direction instead of the element's normal (recall that the interface normal is not aligned with the elements and also that I am only interested in having $[u\cdot n]=0$, whereas the other components can have non-zero jumps) Thanks.

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In the interior of cells, the Raviart-Thomas functions are continuous. As a consequence, the normal component is of course also continuous and the jump is zero. That may not be the case at places where the interface intersects cell boundaries, though. The set of these intersection points, however, consists of a finite number of points along a curve in 2d, or similar in 3d, i.e., a lower-dimensional manifold.

That said, I'm surprised that you need the condition $[u \cdot n]=0$. Typically, one would require that the flux across an interface separating two materials is continuous. The flux is usually a material constant times the vector-valued solution.

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  • $\begingroup$ Yes. $u$ would be my flux in this case. My problem is that my material interface is not explicitly defined. It's defined with a QUAD4 LAGRANGE element that takes 1 or 0 values. In the interface the element will have 0 and 1 as nodal values. This makes the material actually continuous, but with a strong gradient and discontinuous in the limit. $\endgroup$ – balborian May 3 '16 at 14:00
  • $\begingroup$ So in that case, how exactly do you even define what it means for $[u \cdot n]$ to be zero? $\endgroup$ – Wolfgang Bangerth May 3 '16 at 15:05
  • $\begingroup$ The normal can be calculated using several elements. It is true that given that the material field is actually continuous, there are no $[u \cdot n]$ and that I could just have $\nabla \cdot u$ (no body forces in my laplace equation) I wanted to enforce $[u \cdot n]$ because it would be the situation in the limit. Sorry I should have included more details in my question. $\endgroup$ – balborian May 3 '16 at 15:27

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