Lab 05 - Boundary conditions and constraints

Theory and Practice of Finite Elements

Luca Heltai [email protected]

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General Instructions

For each of the point below, extend the Poisson class with functions that perform the indicated tasks, trying to minimize the amount of code you copy and paste, possibly restructuring existing code by adding arguments to existing functions, and generating wrappers similar to the run method (e.g., run_exercise_3).

Once you created a function that performs the given task, add it to the poisson-tester.cc file, and make sure all the exercises are run through the gtest executable, e.g., adding a test for each exercise, as in the following snippet:

TEST_F(PoissonTester, Exercise3) {
   run_exercise_3();
}

By the end of this laboratory, you will have modified your Poisson code to allow also non-homogeneous Neumann boundary conditions on different parts of the domain, and you will have added some more options to the solver, enabling usage of a direct solver, or of some more sofisticated preconditioners.

Lab-05

step-5

1. Add the parameters

   • Neumann boundary condition expression
   • Dirichlet boundary ids
   • Neumann boundary ids

   and the corresponding member variables to your Poisson problem problem, using std::set<dealii::types::boundary_id> for the last two parameters

2. Modify your implementation of Dirichlet boundary conditions, in order to apply the Dirichlet function to all boundary ids indicated in the parameter file

3. Implement Neumann boundary conditions, using the function defined above, on the ids of the Neumann boundary indicated in the parameter file

4. Create a function that parses parameters from a string, to be used in the testing infrastructure

5. Create a few tests that actually solve a Poisson problem on very small grids with very simple but non-trivial combinations of boundary conditions, on different domains, and verify the correctness of your findings. Examples include using globally linear (quadratic) exact functions, with linear (quadratic) finite elements, and verify that the error you make is actually zero (in this case, global interpolation gives the exact solution, therefore the finite element should also provide the exact solution)

6. Modify your code to use AffineConstraints instead of VectorTools::apply_boundary_values (see the documentation of step-6). Run again all tests, and verify that you pass your own checks again with the new code

7. Add the parameters:

   • Local pre-refinement grid size expression

   and the corresponding members. When creating the grid, instead of simply refining globally a fixed number of times (given by the parameter Number of global refinements), refine locally your grid when the function above evaluated in the center of a cell is larger then the actual cell diameter. Make sure you stop refining locally if the number of local refinement cycles you did is equal to the parameter Number of global refinements. Setting the function above to 0 should produce the same results as before, i.e., a fixed number of global refinements up to Number of global refinements.

8. Make sure you compute correctly the hanging node constraints, and that your solver works also with hanging nodes

9. Add a preconditioner to your solver. If you have trilinos installed and it is configured inside deal.II, use its algebraic multigrid preconditioner (it works also with deal.II matrices). Otherwise use one of the other available preconditioners in the library. Verify that your solver is now faster.