ME 540
Numerical Solutions Applied To Heat Transfer And Fluid Mechanics Problems
Krylov and Direct Methods

Part A

Use the program that I have provided (or build it from the source code) to examine the convergence rates of several Krylov methods.  Select a 129x129 array of volumes for the conduction problem, and plot the logarithm of the maximum residual vs.iteration number for the 3 Krylov methods, and the SOR.   You should tell the program not to generate a direct solution.  Accept the default response to questions on the level of incomplete factorization and the number of iterations before restart of GMRES.  Compare run times with those you got for a similar conduction problem with the best combination of options in the MultiGrid program mgsor. Note that comparison of residuals from the MultiGrid and Krylov solvers is a bit like comparing apples and oranges, because the matrices are different as are the forms of the residual used in convergence tests.  I tryed to account for these differences when setting the convergence criteria in the two programs.

Run all methods 5 times to get a feeling for the range of run times.  Construct a table listing the method name, the minimum execution time in your 5 runs, and the difference between maximum and minimum execution times.  Order your table from low to high execution time.  Try to make your computer runs  with as few other applications active as possible.

If you have time, explore the impact of different levels of incomplete factorization and different numbers of iterations before restart of GMRES

Part B

Use the program that I have provided (or build it from the source code) to look at behavior of direct solution for non-diagonally dominant matrices.  Plot the log of maximum absolute error vs. coefficient below the diagonal for  100 equations and coefficient values from 0.0 to 3.0 in increments of 0.2.  Plot the maximum residual for the same set of coefficients.  Aside from incorrect answers, what do you see in the results that should disturb you.

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