Is it possible to use test and trial functions from two different function spaces (defined over two different meshes) in a single weak form? Under what conditions can I do this (eg., each term in the form has functions belonging to only one of the spaces)?

Essentially, I need this to model heat transfer in a block with a channel passing through it - the block is solid and the channel has water flowing through it. The solid domain and fluid domain are two different meshes. Basically I need to implement two equations like

$$ \int_L \mathbf{u}.\nabla T_l \,\tilde{T_l} + \alpha_l\int_L \nabla T_l. \nabla \tilde{T_l} + \frac{\alpha_l}{k_l} \int_{\Gamma_w} h(T_l - T_s) \tilde{T_l} = 0 $$

$$ \alpha_s\int_S \nabla T_s. \nabla \tilde{T_s} + \frac{\alpha_s}{k_s} \int_{\Gamma_w} h(T_s - T_l) \tilde{T_s} - \alpha_s \int_{\Gamma_1} Q \tilde{T_s} = 0 $$ where $T_l$ and $T_s$ are liquid and solid temperatures respectively, $\mathbf{u}$ is the liquid velocity (for which I'm solving a Stokes equation seperately), $L$ and $S$ are liquid and solid domains respectively, $\Gamma_w$ is the wall between the solid and liquid domains and $\Gamma_1$ is a surface of the solid domain where some heat $Q$ is supplied. The water inlet temperature is fixed (a Dirichlet BC).

To have a steady state solution, I need to add these equations to get one. Can anyone tell me how I can go about implementing this in Fenics?

  • $\begingroup$ Look for analyses and implementations of the Petrov-Galerkin method. $\endgroup$ May 20, 2014 at 2:29
  • 1
    $\begingroup$ It sounds like your problem is that you have two regions with different physics interacting. You might want to Google some fluid structure interaction problems in Fenics. $\endgroup$ May 22, 2014 at 8:34
  • $\begingroup$ Thanks for the link, @BiswajitBanerjee . I'll look up Petrov-Galerkin method. $\endgroup$ Jun 3, 2014 at 9:25
  • $\begingroup$ I would like to use a weak formulation which looks like your formulation. Did you found a solution to implement in FENICS? Best Regards, Arnaud $\endgroup$
    – user20345
    May 11, 2016 at 10:19


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.