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| Assignment 1: |
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Choose 3 out of 4 problems for credit. Turn in the solutions to all so I can see where you are.
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Run fem1d.m and confirm using a loglog plot that the convergence
for u(x)=sin(pi*x) in L2 is indeed O(h2).
Now determine the order of convergence in the L∞ norm and
the nodal error l∞. What if your true solution is u=x(1-x)?
- Solve the problem -u''=f with fem1d.m where the exact solution is
u(x)=sin(pi*x)*exp(10*x+1). (You must recompute f(x), of course).
- Use uniform grid with xnel=20,40,80 etc. Confirm the order of convergence with the loglog plot
or calculate the order as discussed in class.
- Use some nonuniform grid as in the following example
fem1d(0,1,[0, .2, .3 ,.7 ,.9]);
(Make sure you're using the most recent version of the code).
By trial and error, choose the optimal placement of nodes to achieve the smallest error
if number of elements = 10. Describe how you did it.
- Find the distributional derivative of u(x)=2x*(1-H(x))+x2H(x).
To which Sobolev space Hk(-1,1) does u belong ? (Find the highest k).
Do the same for u(x)=|x|b and over the interval (0,1). Your answer will depend on b. Repeat over the interval (-1,1).
- Write out the variational and minimization forms (V), (M) for the problem
-u''+u=f, on (0,1), with homogeneous Dirichlet boundary conditions. Write out the FE formulation, and outline the structure of the linear system that needs to be solved.
Extra: implement the corresponding algorithm FE solution in a modified version of fem1d.m (you can find enough there to get started) and test the convergence. Use u(x)=sin(10*x) as the true solution.
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