How to expressive conservative form of pde in ufl

variational formulation
1

I am interested in solving linear and noninear pde in moving domains for which I would to use the conservative form of the pde. For example, the linear heat equation on a moving domain (with material derivative D)

\frac{d}{dt} \int_{\Omega(t)} u \varphi + \int_{\Omega(t)} \nabla u \cdot \nabla \varphi = \int_{\Omega(t)} u D \varphi.

Using back Euler time discretisation and a mesh that moves with a pure Lagrangian velocity (so basis functions have D \varphi_j = 0) we could write this as

\frac{1}{dt} ( \int_{\Omega^n} u^n \varphi_j^n - \int_{\Omega^{n-1}} u^{n-1} \varphi_j^{n-1}) + \int_{\Omega^n} \nabla u^n \cdot \nabla \varphi_j^{n} = 0.

with moving basis functions \varphi_j^n.

To express this in ufl I would need to be able to say that different integrals should be on different meshes. I imagine it would be possible to transfer this to the linear algebra form somehow. Is this possible?

I would also be interested in solving nonlinear pde (e.g. Cahn-Hilliard example https://fenicsproject.org/docs/dolfin/1.4.0/python/demo/documented/cahn-hilliard/python/documentation.html). Could that be possible?

2

I would suggest mapping all integrals to the same domain, using the method of mappings. The material derivative is implemented in UFL, see this paper. This approach is used for shape optimization in dolfin-adjoint, see this paper