The following problem is inspired and directly taken from A preconditioner for solving the inner problem of the p-version of the FEM by Sven Beuchler.
I am trying to find a numerical solution of the following model problem.
$$-\Delta u=f\\u|_{\partial\Omega} = 0$$ in the domain $\Delta =(-1,1)^2$. We can solve the above using the $p-$ version of the FEM with only one element $\Omega$. As the finite element space, we choose
$$M = \{u \in H^1_0(\Omega), u|_\Omega \in P^p\},$$ where $P^p$ is the space of all polynomials of degree $\leq p$ in both variables.
Now, this problem can be discretized as follows. And I need your help in obtaining a geometric visualization of the following problem.
Find $u_p\in M$ such that, $$\int_\Omega \nabla u_p \cdot \nabla v_p d(x,y) = \int_\Omega fv_p d(x,y)$$ for all $v_p \in M$.
I only want a visual interpretation of the above quoted problem considering the finite element space $M$ and it's vectors $v_p$. What does it mean to find a $u_p$ amongst $v_p$s such that the above equation holds? Thank you in advance.