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 
system_matrix(i, j) = (scalar_product(fe_values[a].gradient(i, q_point), 
fe_values[a].gradient(j, q_point)) + scalar_product(fe_values[b].gradient(i, 
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)

system_matrix(i, j) += (scalar_product(fe_values[b].gradient(i, q_point), 
fe_values[a].gradient(j, q_point)))*fe_values.JxW(q_point);
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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