Lab 06 - A posteriori error estimation and adaptive FEM

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-06

step-6

1. See documentation of step-6 at https://www.dealii.org/current/doxygen/deal.II/step_6.html

2. Add the parameters

   • Mapping degree
   • Marking strategy
   • Estimator type
   • Coarsening and refinement factors

where Mapping degree controls the degree of the mapping used in the code, Marking strategy is a choice between global|fixed_fraction|fixed_number, Estimator type is a choice between exact|kelly|residual, and Coarsening and refinement factors is a std::pair<double, double> containing the arguments to pass to the GridRefinement::refine_and_coarsen_fixed_* functions

1. Make sure all FEValues classes use a mapping with the correct order, and make sure you use the correct mapping in the output as well (if you have a recent Paraview)

2. Add a Vector<float field error_per_cell to Poisson, to be filled by the method estimate

3. Add a method residual_error_estimator that computes the residual error estimator, using FEInterfaceValues and FEValues

4. Add a method estimate to the Poisson class, to compute the H1 seminorm of the difference between the exact and computed solution if Estimator type is exact, calls KellyErrorEstimator<dim>::estimate if Estimator type is kelly, and calls residual_error_estimator if Estimator type is residual

5. Add to the convergence tables also the estimator you computed. This should be identical to the H1 semi-norm in the case where Estimator type is exact

6. Taking the exact solution computed on the L-shaped domain, compute the rate at which the adaptive finite element method converges in terms of the number of degrees of freedom using the three estimators above

7. Set zero boundary conditions, forcing term equal to four, exact solution equal to -x^2-y^2+1 and solve the problem on a circle with center in the origin and radius one, for various finite element and mapping degrees. What do you observe when the mapping degree does not match the finite element degree? How do you explain this?

8. Create a test that reproduces exactly the behaviour of step-6, using only your parameter file