Lab: Adaptive mesh refinement

Material 

1. Lecture notes: Chapters 1-5 (specifically Section 3.2)

2. Template file from GitHub: template-report-Stokes-AMR.ipynb

3. For extra assignment: 

Hoffman, Johan, et al. "Towards a parameter-free method for high reynolds number turbulent flow simulation based on adaptive finite element approximation." Computer Methods in Applied Mechanics and Engineering 288 (2015): 60-74. Links to an external site.

Mandatory assignment (1p if submitted in time for deadline)

1. The domain

Use the data to the adjoint problem which defines the functional equal to the drag force:

psi_1 = (0,0), psi_2 = 0 and psi_3 = (1,0)

Then change the height of the domain to H=L, and move the center of the circle over the domain. Describe how the adjoint solution and the local mesh refinement changes as a result. 

2. The functional 

Change the data to the adjoint problem (psi_1, psi_2, psi_3) and describe how the adjoint solution and the local mesh refinement changes as a result. 

Extra assignment (choose one: 1p if submitted in time for deadline)

1. The total error 

With the same approximation spaces for the primal and the adjoint problem, note that the total error appears to be close to zero. This is a false conclusion, and a consequence of the fact that we approximate the exact adjoint solution with a finite element function in the test space of the primal problem, for which the weak form is zero (modulo errors from solving the algebraic system). Change the mixed finite element space of the adjoint equation such that the total error is not zero, for example, increase the polynomial order of the adjoint finite element spaces. Then verify that the total error is reduced when you use a mesh with finer resolution.  

2. Adaptive algorithm

Extend the template file into an adaptive method for recursive local mesh refinement based on the error indicator from the adjoint solution, with a stopping criterion based on the total error.  

3. Navier-Stokes equations 

Extend the template file to the Navier-Stokes equations, for which you solve the adjoint Navier-Stokes equations by time stepping backwards in time from the end time to the start time, linearised at the primal solution.