# Why lattice Boltzmann despite its huge number of mesh points still has worse accuracy in comparison to FEM for calculating wall shear stress?

I'm just doing a very simple experiment. I'm calculating wall shear stress based on Poiseuille flow for a pipe by using lattice Boltzmann method (LBM) and FEM to compare their values with the analytical solution, which is calculated as:

$$\tau = \frac{2 \mu u_\max}{R}$$

Where we have: $$u(r) = u_\max \left(1 - (\frac{r}{R})^{2}\right)$$ and $$\tau = - \mu \frac{\partial u}{\partial r}|_{r=R}$$.

For a pipe with $$R = 10$$mm and $$L=100$$mm for its radius and length as well as $$\mu = 0.004$$ $$\mathrm{Pa}\cdot\mathrm{s}$$ and $$u_\max = 0.0125$$ $$\frac{\mathrm{m}}{\mathrm{s}}$$:

$$\tau = \frac{2 \times 0.004 \times 0.0125}{0.01} = 0.01$$

So: $$\tau = 0.01$$ Pa.

I did the simulation with LBM with a resolution of $$0.16$$mm and I got the value: $$\tau_\text{LBM} = 0.010597292391$$ Pa.

On the other hand, I did the simulation with FEM with a resolution of $$2$$mm and I got: $$\tau_\text{FEM} = 0.0097797$$ Pa.

You see that the error of FEM is around $$2.2$$%, but the error of LBM is around $$6$$%, despite a factor of magnitude coarser resolution of FEM!

For those of you that are familiar with LBM: this LBM simulation is done by using D3Q27 lattice and BFL boundary condition. When I used a simple bounce back instead of BFL, I got $$\tau_\text{LBM} = 0.0089005915558$$ Pa, which its error is around $$11$$%.

My main application for using LBM is for a really sensitive biofluidic framework to simulate blood flow in brain arteries. If LBM fails to calculate wall shear stress accurately even in this simple situation of a pipe with Poiseuille flow, how can I trust it to use it for much more complex geometries and flow conditions of blood flow in brain vessels? Why LBM despite its much finer resolution still falls behind the FEM even with a factor of magnitude coarser mesh size? I appreciate any hint or suggestion.

• Does FEM that you use have hp-refinement enabled? what order of elements is used in FEM? (I am coming from a very different application area, so forgive me if my questions do not apply to the problem you are solving) – Anton Menshov Feb 14 at 23:19
• @AntonMenshov No there is no refinement in FEM framework that I used. Elements are just P1-P1 for velocity and pressure. – Alone Programmer Feb 14 at 23:26