Dear all, I want to solve a linear order PDE system:
\begin{align}
\left\{
\begin{array}{rcll}
\mathrm{i} \beta \partial_x \psi + \mathrm{i} \alpha \partial_y \psi + k_a\left[n_a(x, y) - 1\right]I \psi &=& 0, & \text{in } \Omega, \\
\psi(x, y) &=& g(x, y), & \text{on } \partial \Omega.
\end{array}
\right.
\end{align}
where
\begin{equation}
\alpha = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}, \quad
\beta = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix},
\end{equation}
and \psi=(\psi_1,\psi_2) is a linear system
During my derivation of the weak form, I find that I need to use the vector value test function v=(v_1,v_2). Besides, terms like \partial_xv_1, \partial_y v_2 will appear too. My questions are that how to write such weak form in dolfinx and how to describe terms like \partial_xv_1, \partial_y v_2?
Thanks for your advice. I want to write it in Dolfinx. So how to interpolate this quantity into a suitable space? Also I want to use integration by parts, so my goal is to use the derivative of the test functions like v_{1,x}, since I will use a vector test functions.
You can use u.dx(i)
in DOLFINx as well as legacy dolfin.
Interpolation of this quantity can be done with DOLFINx.fem.Expression: