It is about the implementation of fictitious domain method (FDM) in FEniCS.
We have two domains \Omega_1, \Omega_2, where \Omega_1 is a background domain and \Omega_2 is a small domain. A flow mapping \mathbf{X} : \Omega_2\to \tilde{\Omega}_2\subset \Omega_1 maps \Omega_2 to a subdomain in \Omega_1.
Let \lambda \in H^1(\Omega_2) and v\in H^1(\Omega_1). How to assemble the following linear form
\int_{\Omega_2} \lambda(\hat{x}) v\left(\mathbf{X}(\hat{x})\right)~d\hat{x}
where v(x) is a testing function defined on CG-P1 space on \Omega_1, \lambda(\hat{x}) is a coefficient function, and \mathbf{X}(\hat{x}) is a vector CG-P1 function.