In this program examples from TouWi17 (SISC) https://epubs.siam.org/doi/abs/10.1137/16M1067792 can be computed with a Newton-Krylov (CG) solver and geometric multigrid preconditioning. The program is based and tested on deal.II version 9.6.0 (2024-2026) and 9.7.0 (from Aug 2026). Building and running the program is a usual when working with deal.II tutorial steps: cmake (to build once the Makefile), afterwards make run, i.e., make debug run or make release run.
The base program uses p=2, which corresponds to the classical Poisson problem: Find u:\Omega\to R such that -\Delta u=f in \Omega; here on the unit square \Omega=(0,1)^2, and homogeneous Dirichlet boundary conditions, and right hand side f=-1 . Changing test cases is done towards the end after `// Defining test cases'.
More information in terms of a bullet list are:
- 3 test cases; grep for "Defining test cases"
- The p (of p-Laplacian) can be changed in power_p in set_runtime_parameters... of each respective test case
- A geometric multigrid method
- But also switched (via "linear_solver_type") to a LU method (UMFPACK)
- The nonlinear problem is solved with Newton's method
Two Newton methods are implemented:
- a) a standard line search (see paper TouWi17, SISC, 2017)
- b) a modified Newton method in which certain terms are scaled
- Evaluation of several norms. The most important is the F-norm that is equivalent to the H1 norm for the case p=2
- The errors and also solvers performances can be grepped from the terminal output via grep "Errors" file_name_terminal.txt or grep "Info" file_name_terminal.txt . One suggestion to run [Terminal>] make release run | tee results.txt . Then, [Terminal>] grep "Errors" results.txt and [Terminal>] grep "Info" results.txt
- Graphical output is written in *.vtk that can be displaced by paraview or visit.
- Simulations of TouWi17 using this Newton-Krylov GMG method can be found in Wi22_NumPDE https://repo.uni-hannover.de/items/33ce4843-9a8d-480f-ad38-2c0bd0f9cae5 p. 260: Section 10.8 p. 333ff : Section 13.15
Contact: in case of questions, do not hesitate to contact me via email: thomas.wick@ifam.uni-hannover.de