Why is the continuous Galerkin Finite Element Method a poor choice for the inverse problem for the Navier-Lame equation?

Question

I work in a field (elasticity reconstruction) that frequently uses standard Finite Element methods to solve the first order PDE, where we are given u and solve for mu and lambda:

$$ (\mu(u_{i,j} + u_{j,i}))_{,j} + (\lambda~div~\mathbf{u})_{,i} = -\rho \omega^2\mathbf{u}_{i} $$

known as the Navier-Lame equation. Here we try to solve the "inverse" case (given u solve for mu and lambda) which is a first-order PDE (while the equation is more commonly solved for u which is a second order PDE).

The results overall have been a bit underwelming. Over the past year I was able to speak briefly about this problem to four quite prominent applied mathematicians. Their consensus was that standard FEM, while appropriate for elliptic PDEs of second order, is an inappropriate and unstable mathematical basis for a first order PDE, while better solutions might be Finite Volume or discontinuous Galerkin.

All I remember of the discussions is that first order PDEs have characteristic curves and can handle jumps, while finite elements, like second order PDEs, are smooth in C1 throughout.

I cannot find any supporting information for this viewpoint in a text. Can anyone explain why standard FEM is a poor choice for a first order PDE? Or is this not right?

Answer 1

The issue with finite elements in the current context is not that you have a first order differential equation, but with the kind of first order equation you have. In general, finite element methods can be made to perform quite well for first order problems such as advection equations. But that's beside the point here.

To illustrate the issue, think of the simpler problem of identifying the coefficient $a(x)$ from the equation $$ -\nabla \cdot (a(x) \nabla u(x)) = f(x) $$ assuming that you know $f(x)$ and $u(x)$. It is true that you can write this as a first order equation for the unknown $a(x)$: $$ \beta \cdot \nabla a + \gamma a = f(x) $$ where $\beta(x) = -\nabla u(x)$ looks a lot like an advection direction, and $\gamma(x)=-\Delta u(x)$ like a reaction coefficient. So this looks like an advection-reaction equation.

But the solution of this problem is not well posed unless (i) $|\beta|\ge \beta_\text{min}>0$, i.e., the "wind speed" is always bounded away from zero, (ii) unless each point $x$ in the domain is connected by a "streamline" of the vector field $\beta$ to the boundary, and (iii) you have boundary values for $a(x)$ at all inflow boundary parts, i.e., where $\beta\cdot n<0$.

But, in general cases, none of these three conditions are satisfied. You can easily see this if you consider a circular domain and you have $f=1$ in the outer half of the circle and $f=-1$ in the inner half (and let's assume $a=1$ everywhere). Then your solution $u(x)$ will be rotationally symmetric, and from the boundary first rise up and the decrease again -- i.e., you'll have a circular ring. In this case, each point outside the rim of the hill is connected to the boundary by a streamline, but not on the inside of the hill. Furthermore, $\nabla u=0$ along the rim of the circular hill.

In situations like this, the first-order PDE is simply not a well posed problem for $a(x)$. No method will be able to yield a satisfactory solution in such cases, and that includes the finite element method.