<jupyter_start><jupyter_text>Fast PDE/IE course, Skoltech, Spring 2015  Midterm study guideFor the midterm you should be able to:1. Finite differences (FD)    - write down basic FD approximations to differential operators, equations, and boundary/initial conditions    - determine the order of approximation (and prove)    - judge whether an initial-value problem is stable (or for what parameters it is stable)    - understand how to implement FD on a computer2. Finite elements (FE)    - fomulate a (boundary-value problem for a) PDE in a weak form    - formulate a FE problem, finite element space (piecewise linear functions)    - write down the stiffness matrix in simple cases    - understand how to implement FE on a computer3. Spectral methods (SM)    - formulate a spectral method for a given PDE    - write down the stiffness matrix in simple cases    - understand how to implement SM on a computer4. Multigrid    - understand the idea of multigrid: smoothing, restriction, prolongation    - understand how to choose the $A_h$ operators on a coarse grid from a fine grid     - formulate the smoothing property in terms of high/low frequences<jupyter_code>from IPython.core.display import HTML
def css_styling():
    styles = open("./styles/alex.css", "r").read()
    return HTML(styles)
css_styling()<jupyter_output><empty_output>
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