Harj. 5

25.3. -99

1. Download fem1.mws . Modify the example (from [EEHJ] p. 120) to work correctly.

   -u''=f  in (0,1)
   u(0)=u(1)=0,
   f(x)=10(1-10(x-0.5)2)e-5(x-0.5)2
   

2. See the Matlab-file fem1mlb.html which includes a problem from [T-W] and its FD solution + an untested beginning of FEM-solution. Complete, solve and compare. (You may very well create your own style of solution, both in 1) and 2).
   Do also problem 1) with Matlab.
3. • (a) Write a Matlab m-file that solves a Sturm-Liouville problem of the form

           -(a(x)u'(x))' = f(x), 0 < x < 1
            Dirichlet- or Neumann- or combination bdary cond.
     

     Make your program handle all combinatios of Dirichlet/Neumann (homog. or non- homog.) (even Robin if you wish) conditions. Also let your mesh be any (increasing) sequence of points on [0,1]. Use numerical integration (Trapezoidal rule or Gaussian mid-point rule) in assembling the RHS. The quadrature rule should be at least the same order as the error behaviour of the method without numerical quadrature, we expect the error to be the same order as in linear interpolation, that is
     O(h2)
   • (b) Use your program especially on [T-W] Exe 2.14, p. 76 (Exercise set 3, problem nr. 3) and include a similar analysis of convergence and error behaviour as in 2.14.
4. Solve some of the Dirichlet problems in exercises nr. 4 using PDE Toolbox. Make a comparison to exact sol or to the FFT-based solution provided by Cooper's dirch . Evaluate max error (at grid points or at interpolated mesh points) and see how it is affected by mesh refinement. Make graphic illustrations.