# Finite element: handle discontinuity/abrupt interface

I am using Galerkin's method to set up some pde solver for my problems(1D/2D) at hand.

Some abrupt material interfaces do exist, so far, what I did regarding this is simply:

1) no element across the interface, i.e. nodes exactly lie at the interfaces.

2) no special B.C. imposed on interfaces, just natural B.C.(as it is).

I am wondering:

1) is what I did sufficient to deal with the interfaces?

2) some people recommend to use zero thickness elements along the interfaces, so that every nodal point on the interface is associated with a duplicate node, is it necessary? and what is the justification?

It depends on the physics of your problem. Let us take a second-order elliptic equation $\nabla\cdot(a\nabla u) + k^2 u = f$ for example, where $a$ and $k$ are discontinuous across interface.

In most common case, across the interface you have $u$ and $(a\nabla u)\cdot \mathbf{n}$ matched from two sides of the interface, so that your do-nothing and assembly (unique dofs on interface) strategy is conforming to the physics.

But there are some cases, the physics is different. For example, in electromagnetic, there might be surface (interface) charges, so that for transversal field you get a jump $[(a\nabla u)\cdot \mathbf{n}]=S$ where $[v]=v_1-v_2$ and $v_i$ is the trace from $\Omega_i$ for the interface between $\Omega_1$ and $\Omega_2$ while $u$ keep continuous across interface.

Even weird, for field component normal to interface, you get the different things: $[\nabla u\cdot \mathbf{n}]=S$ and $[ku]=0.$ These are very unusual. So it is not the mathematics but the physics one should ask for.

• for the moment I do not consider the transversal field in my problem. what if there is an additional first order convection term in the elliptic PDE? is the do-nothing strategy still justified? – lorniper Jun 1 '14 at 7:50
• @lorniper: Please write down your physics in conservation form (i.e. integral form). – Hui Zhang Jun 1 '14 at 11:13
• well, e.g. a standard steady-state convection diffution reaction problem? – lorniper Jun 2 '14 at 10:46
• @lorniper I would say in most common case, convection and reaction makes no difference to the diffusion problem, that is, flux and Dirichlet trace are continuous across the interface. But you really need to pay attention to your physical model. Maybe you have interface sources that give jumps, who knows?! – Hui Zhang Jun 2 '14 at 12:54

For a material property change for elliptic PDEs you shouldn't have to do anything special. It's best to put the nodes on the interface, but you shouldn't even need to impose a BC. It's not a boundary. Just a change in the coefficient.

• yes, i didn't put any B.C. just assemble as in the homogenous case. I wonder in which cases this needs some more fix, hyperbolic pde? – lorniper May 29 '14 at 21:33
• I'm a little far from the literature these days, but I know that material property discontinuities can cause reflections of waves and other phenomena that you might want to look more carefully at. I think the mathematics are similar, but if capturing these phenomena is important to your work, you may want to use special methods. – Bill Barth May 30 '14 at 12:35
• do you have some key words in mind? so i can have a simple check. – lorniper May 30 '14 at 12:38