# [deal.II] Setting off-diagonal elements in the matrix

Having a coupled set of poisson-like equations
\partial_t a = \nabla^2 a
\partial_t b = \nabla^2 b + \nabla^2 a
I can write the weak form(s) as (while ignoring boundary conditions)
\partial_t a = (\nabla v_1, \nabla a)
\partial_t b = (\nabla v_2, \nabla b) + (\nabla v_2, \nabla a)
Now in deal.II I add both equations to the same matrix (first the main
diagonals):
q_point), fe_values[b].gradient(j, q_point)))*fe_values.JxW(q_point);

and then the off-diagonal term
\partial_t b = (\nabla v_2, \nabla a)

But wouldn't that be the same as for the system of equations
\partial_t a = \nabla^2 a + \nabla^2 b
\partial_t b = \nabla^2 b
?
How does deal.II distinguish between both?

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