I'm trying to figure out of something from Claes Johnson's "Numerical solution of partial differential equations by the finite element method" Chapter 11.
Given the problem: Find $(u_h,p_h)\in V_h\times H_h$ such that
$$ (\nabla u_h,\nabla v)-(p_h,\operatorname{div}v)=(f,v)\quad\forall v\in V_h\\ (q,\operatorname{div}u_h)=0\quad\forall q\in H_h $$ where all inner products are $L_2$-inner products. Taking $v=u_h$ and $q=p_h$ and adding them together I get $$ (\nabla u_h,\nabla u_h)=(f,u_h) $$ and Johnson claim I should get $$ ||u_h||_1^2=(f,u_h) $$ where $$ ||w||_1^2=\sum_i\int_\limits\Omega(|w_i|^2+|\nabla w_i|² )\,\mathrm{d}x $$ However, since the inner products are $L_2$-inner products, I do not see from where I should get the first term in the integral. Any guidance appreciated.