Finite element method in $x$, $y$, spectral method in $z$

Question

I'm working on a certain problem of slow, non-Newtonian, thin-film flow. This problem can be modelled with the incompressible Stokes equations:

$\nabla\cdot 2\mu(\dot\varepsilon)\dot\varepsilon - \nabla p + \rho g = 0, \qquad \nabla\cdot u = 0$

where $u$ is the velocity, $\mu$ is the (nonlinear) viscosity, $\dot\varepsilon$ is the strain rate tensor, $p$ the pressure, $\rho$ the fluid density and $g$ the gravitational acceleration vector. In my field, there's a good approximation for when the fluid doesn't slip at all along the lower boundary, and a different approximation for when the fluid is in free slip along the lower boundary. The Stokes equations are costly to solve in the whole domain, so some folks have tried to come up with approximations to it which work in both the no-slip and free-slip regimes.

My idea to approach this is to use semi-discretization in the $z$-direction, but with spectral methods. From looking at experimental evidence, I have a strong hunch that most of the vertical variation could be captured with the Legendre polynomials up to degree 4. The velocity field would be approximated as

$u(x, y, z) = \sum_lp_l(z)u_l(x, y)$

to derive a coupled system of PDE for the fields $u_l$. This coupled system for $\{u_l\}$ can then be solved using, e.g. Galerkin finite element methods, finite difference methods, etc.

I'm looking for any references or case studies where this approach has been used. I googled and found this paper on solving the Navier-Stokes equations in domains with only one direction of periodicity. There's also this paper which is more from a model reduction point of view. If there's a buzzword for this kind of approach I'd like to know what it is.

Finally, any software implementations would be helpful. I can write this from the ground up using deal.II myself. I'm less familiar with FEniCS, Firedrake, or LifeV; do any of them have a way to create Frankenstein finite element spaces like this?

Answer 1

If I understood your problem correctly, this paper may help you. In this paper, they are trying to solve a thermo-mechanical coupled problem. The structural equations in the domain are modeled using 1-D models and thermal equations in the domain are modeled using 2-D. Therefore they have consistently reduced the dimension of thermal equations in 2 D by using a selective weighted residual technique in one direction which render it 1-D. They approximate it as $T(x,z,t)=g(z)\bar{T}(x,t)$. please refer to the paper for detailed implementation. Hope this will help youLink to the paper http://www.sciencedirect.com/science/article/pii/S0020740315000600

Answer 2

Not a Franken-element at all :-) If you know the number $N$ of modes you want to consider in z-direction, then you simply get a coupled system of $N$ dim-1 dimensional PDEs, all of which happen to have fundamentally the same structure.

The place where this kind of discretization is most often used is if you have a cylindrical coordinate system. There, people often use a Fourier transform in the angle $\varphi$. If you keep the first $N$ modes, then you end up with a system of $N$ couples PDEs in $r$-$z$ only. I've seen this many times, but can't recall any concrete papers other than the ones by J.-L. Guermond on magnetohydrodynamics. His whole Sfemans code is based on this concept.