Restricted to one dimensional problem, is it possible to dynamically adapt the finite element method (FEM) discretisation based on the local value of the Péclet number ($P_e$) for advection-diffusion problems?
For advection-diffusion problems central discretisation becomes unstable when $P_e>2$, that is, when the problem becomes advection dominated.
A common way to solve this problem for the finite volume method (FVM) is with exponential fitting. For example, when solving the advection-diffusion equation the diffusion term is discretised with a central difference stencil, $$d(x)u_{xx} \approx \boldsymbol{D}_2\boldsymbol{u}(u_{j-1}, u_j, u_{j+1})$$
The advection term is discretised as a weighted sum of a central difference and a upwind stencil, $$a(x)u_x \approx \kappa \boldsymbol{D}_1\boldsymbol{u}(u_{j-1}, u_j, u_{j+1}) + (1-\kappa) \boldsymbol{U}_1 \boldsymbol{u}(u_j, u_{j-1})$$
NB There are much easier ways to incude exponential fitting with the FVM, the above illustrates the concept. To see how it is done in practice I refer you to my notes, http://danieljfarrell.github.io/FVM/advection_diffusion.html#adaptive-upwinding-exponential-fitting.
This nice feature about exponential fitting is that it chooses the value of $\kappa$ such that the discretisation weights in favour of the stable upwind stencil when $P_e$ is large and in favour or the higher accuracy central stencil when $P_e$ is small. This is done locally based on the value of $P_e$. It also has the nice property of preserving monotonicity of the solution (at least in 1D) which is a nice feature for physical simulations (conservation laws).
How can exponential fitting be introduced with the FEM in 1D? Does the approach also preserve monotonicity as with the FVM?
