MA 3065 Final: hw5 (1)

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Math 8590 – Homework 5 (Due Friday Nov 16)
Please hand in your solution to 1 problem from those below.
1. Prove the O(h)convergence rate for monotone finite difference scheme using inf- and
sup-convolutions, instead of the doubling variables argument. Your proof should be
similar to the alternative proof of the O(ε)rate in vanishing viscosity (from Section 8
in class notes).
2. Write code in your favorite programming language to solve the shape from shading
problem in n= 1 dimension. You can use fast marching or fast sweeping to solve the
eikonal equation, and decide on an appropriate boundary condition.
3. Write code in your favorite programming language to simulate an n= 1 dimensional
homogenization problem of the form
uε+|u
ε(x)|=fx
ε,
on the domain (0,1) with homogeneous Dirichlet conditions, where fis 1-periodic. Solve
the rapidly oscillating equation for uε. Try to find nonconstant ffor which you can solve
the cell problems and compute explicitly H(p). Then solve the effective equation for u,
comparing results for small ε.
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