Do the following two forms have the same meaning

Question

Hello,

Say I want to write $a = K*u_0$ , where $K$ is stiffness matrix and $u_0$ is initial condition.

So are the following two forms the same?

k = inner(nabla_grad(u),nabla_grad(v))*dx # stiffness matrix integrand
K = assemble(k) 
a = K*u_0

OR

a = inner(nabla_grad(u_0), nabla_grad(v))*dx

Answer 1

These are the same (up to boundary conditions).
It follows from the finite element method. If you
set up
$u = \sum_i u_i N_i$
$u_0$ is the vector containing $u_i$,
$N_i$ is trial function and $N_j$ is test function then you
may prove it by writing out the expressions for K and a.