The steps include: writing the variational formulation, gridding the domain, and coding the appropriate functions for the functions f, k, and u_d.

• Please share with me a file in CANVAS with the final report.
• Your files should be called MTH-655-LAB5_Person1_Person2_.....REPORT.pdf
  and
  MTH-655-LAB5_Person1_Person2_.....JOURNAL
  (Journal is not mandatory)
• Solve Parts 2 and 3. Or, solve Parts 3 and 4.
• Group size can be up to 6 for this assignment, as long as you solve the extra parts (For smaller groups, please discuss with me how much exactly you are expected to do). A large group should have a manager who will do most of the delegating and reporting, rather than experiments or coding. In your report please describe the roles/contributions of each group member. Make sure that everyone understands each part of the solution.
• I can help you to move forward by communication through CANVAS. I will respond by making notes in the journal.
• If you choose to use FE model other than the triangular grid in ACF, my templates will likely not be directly useable. As part of your solution, please provide an explanation what you had to do to complete the assignment.
• THE REPORT IS DUE BY MONDAY 3/6

1. Manufacture f and u_d for ACF so that the true solution is

   u(x,y)=1-x² +e^3y.

   Code the true solution in a function exact.m (You can use it of course also for u_d).
   

   Hint:Your solution can be coded like this (adapted from f.m)
   function uexact = exact(x)
   %UEXACT
   uexact = x(:,1)+3*exp(x(:,2));
   

   
   Run the code and plot both the true solution and the numerical solution to check for obvious mistakes.
2. Now calculate the actual H1 and L2 errors for the problem (a). You may use the provided functions basisfun.m, solerr.m, and tri_quadcofs.m. Follow the template of error calculations and instructions in solerr.m.

Report on the convergence of the solutions to Part 1 using a sequence of meshes with different h

Consider now the stationary heat conduction problem with a true solution as in Part 1, but with the coefficient

k(x,y)=x+y+A.

1. Make the appropriate changes to the ACF code to solve this problem. (Show me only the parts that you modified). Report on convergence as in Part 2.
2. Extra: make the coefficient discontinuous so it is equal 1 everywhere except in the circles as in lab 3, where it equals 10. Of course now you do not know the true solution.
3. Extra: Experiment with A for which there is no coercivity. What happens?

The code described below provides an estimate on local H1 error of the solution, and thus it can be used to adapt the grid.

[uest] = esterr(elements3,coordinates,full(u));
[v,j]=max(sqrt(uest));
%%%% Display the local error by assigning it to the nodes %%%% (display the estimated error)
uloc = zeros(size(coordinates,1),1);
for j = 1:size(elements3,1)
uloc (elements3(j,:)) = uloc(elements3(j,:)) + (uest(j))/3;
end
show(elements3,elements4,coordinates,uloc(:));
In practice one marks the elements which need refinement rather than the nodes. We choose the latter option here to not complicate things too much.

• Apply your estimator to the problem in Part 1a, or Part 3a, or Part 3b, or to all for extra credit.
• Extra: If you know your true solution, you can check how well the local estimator marks the regions that need the refinement. You can also calculate the local H01 error and compare to that provided by the estimator.
• Extra: Refine the grid where the indicator tells you and check if the error has decreased.