I want to solve the Navier-Stokes equations numerically. After choosing a temporal discretization scheme and a finite element for the space discretizaion, I need to deal with a system of linear equations.
Suppose we're using Taylor-Hood finite elements $\mathbb P_2/\mathbb P_1$, i.e. the finite elements on which the velocity approximation is searched consists of piecewise quadratic polynomials.
One of the constituents of the coefficient matrix of the linear system involves integrals of the form $$\int_\Lambda(u^0\cdot\nabla\psi_i)\psi_j\:{\rm d}\lambda$$ where $u^0$ is the velocity approximation from the last time step and the $\psi_i$ and $\psi_j$ are two of the local degrees of freedom (which are quadratic polynomials here) of the reference element, which I have chosen to be the triangle $\Delta$ spanned by $(0,0)$, $(1,0)$ and $(0,1)$.
The presence of $u^0$ forces me to use quadrature for the approximation of the integral. Suppose we're using Gaussian quadrature as described in my other question.
$u^0$ contains the values of the velocity approximation from the last time step at the nodes of the finite element space. Using quadratic Lagrange elements, the local nodes are the vertices of the triangles together with the midpoints of the edges. Now, if I want to use quadrature, didn't I need to choose the sampling points to be precisely the local nodes of the reference element?
Moreover, I would like to now if there is anything else I need to take care of.