COMSOL 6.2 - References for the Linear System Solvers and the Preconditioners

8. D. Kuzmin, R. Löhner, and S. Turek (editors), “Flux-Corrected Transport: Principles, Algorithms, and Applications,” Scientific Computation, Springer-Verlag, Berlin, 2005.

9. A. Greenbaum, “Iterative Methods for Linear Systems,” Front. Appl. Math., vol. 17, SIAM, 1997.

10. Y. Saad and M.H. Schultz, “GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems”, SIAM J. Sci. Statist. Comput., vol. 7, pp. 856–869, 1986.

11. M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, and S. Maiti, “Recycling Krylov Subspaces of Linear Systems,” Siam J. Sci. Comput., vol. 28, no. 5, 2006, https://epubs.siam.org/doi/10.1137/040607277.

12. Y. Saad, Iterative Methods for Sparse Linear Systems, Boston, 1996.

13. H. De Sterck, U.M. Yang, and J. Heys, “Reducing Complexity in Parallel Algebraic Multigrid Preconditioners”, SIAM J. Matrix Anal. Appl., vol. 27, no.4, 2006.

14. B. Metsch, Algebraic Multigrid (AMG) for Saddle Point-Systems, PhD thesis, Bonn, 2013.

15. U. Meier Yang, Parallel Algebraic Multigrid Methods — High Performance Preconditioners, LLNL.

16. H. De Sterck, R.D. Falgout, J. Nolting, and U. Meier Yang, “Distance-two interpolation for parallel algebraic multigrid,” Numer. Linear Algebra Appl., vol. 15, no. 2–3, pp. 115–139, 2008.

17. Y. Saad, “A Flexible Inner-Outer Preconditioned GMRES Algorithm,” SIAM J. Sci. Statist. Comput., vol. 14, pp. 461–469, 1993.

18. M.R. Hestenes and E. Stiefel, “Methods of Conjugate Gradients for Solving Linear Systems,” J. Res. Nat. Bur. Standards, vol. 49, pp. 409–435, 1952.

19. C. Lanczos, “Solutions of Linear Equations by Minimized Iterations,” J. Res. Nat. Bur. Standards, vol. 49, pp. 33–53, 1952.

20. H.A. Van Der Vorst, “A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems,” SIAM J. Sci. Statist. Comput., vol. 13, pp. 631–644, 1992.

21. J.R. Gilbert and S. Toledo, “An Assessment of Incomplete-LU Preconditioners for Nonsymmetric Linear Systems,” Informatica, vol. 24, pp. 409–425, 2000.

22. Y. Saad, ILUT: A Dual Threshold Incomplete LU Factorization, Report umsi-92-38, Computer Science Department, University of Minnesota, available from www-users.cse.umn.edu/~saad/.

23. W. Hackbusch, Multi-grid Methods and Applications, Springer-Verlag, Berlin, 1985.

24. R. Beck and R. Hiptmair, “Multilevel Solution of the Time-harmonic Maxwell’s Equations Based on Edge Elements”, Int. J. Num. Meth. Engr., vol. 45, pp. 901–920, 1999.

25. S. Vanka, “Block-implicit Multigrid Calculation of Two-dimensional Recirculating Flows,” Comput. Methods Appl. Mech. Eng., vol. 59, no. 1, pp. 29–48, 1986.

26. H.C. Elman and others, “A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations,”, SIAM J. Sci. Comp., vol. 23, pp. 1291–1315, 2001.

27. A. Toselli and O. Widlund, “Domain Decomposition Methods — Algorithms and Theory,” Springer Series in Computational Mathematics, vol. 34, 2005.

28. E. Agullo, L. Giraud, A. Guermouche, A. Haidar, and J. Roman, Parallel algebraic domain decomposition solver for the solution of augmented systems, HAL Id: inria-00559133, inria.hal.science/inria-00559133.

29. A. Haidar, On the parallel scalability of hybrid solvers for large 3D problems, PhD dissertation, INPT, June 2008, TH/PA/08/57, oatao.univ-toulouse.fr/7711/

30. X. Antoine and C Geuzaine, “Optimized Schwarz Domain Decomposition Methods For Scalar and Vector Helmholtz Equations,” Modern Solvers for Helmholtz Problems, Birkhäuser, 2017.

31. J.E. Dennis, Jr., and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM, 1996.

32. Intel® Developer Zone, Preconditioners based on Incomplete LU Factorization Technique: https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-c/2023-1/precondition-based-on-incomplete-lu-factorization.html.

33. R. Fruend, “A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems,” SIAM J. Sci. Comput., vol. 14, pp. 470–482, 1993.

34. H.M Bücker, A transpose-free 1-norm quasi-minimal residual algorithm for non-Hermitian linear systems, FZJ-ZAM-IB-9706.

The COMSOL Multiphysics implementations of the algebraic multigrid solver and preconditioner are based on the following references:

35. K. Stüben, Algebraic Multigrid (AMG): An Introduction with Applications, GMD Report 70, GMD, 1999.

36. C. Wagner, Introduction to Algebraic Multigrid, course notes, University of Heidelberg, 1999.

37. R. Hiptmair, “Multigrid Method for Maxwell’s Equations,” SIAM J. Numer. Anal., vol. 36, pp. 204–225, 1999.

38. D.J. Mavriplis, “Directional Agglomeration Multigrid Techniques for High-Reynolds Number Viscous Flows,” ICASE Report No. 98-7 (NASA/CR-1998-206911), Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, VA, 1998.

39. V. John and G. Matthies, “Higher-order Finite Element Discretization in a Benchmark Problem for Incompressible Flows,” Int. J. Numer. Meth. Fluids, vol. 37, pp. 885–903, 2001.

40. V. John, “Higher-order Finite Element Methods and Multigrid Solvers in a Benchmark Problem for the 3D Navier–Stokes Equations,” Int. J. Numer. Meth. Fluids, vol. 40, pp. 775–798, 2002.

41. J. Zhu, X. Zhong, C.-W. Shu, and J. Qiu, “Runge–Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes,” J. Comput. Phys., vol. 248, pp. 200–220, 2013.

42. X. Zhang, Y. Xia, and C.-W. Shu, “Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes,” J. Sci. Comput., vol. 50, pp. 29–62, 2012.

43. AMS — hypre 2.23.0 documentation, https://hypre.readthedocs.io/en/latest/solvers-ams.html.

44. R.R. Craig and M.C.C. Bampton. “Coupling of Substructures for Dynamic Analyses,” AIAA J., vol. 6, no. 7, pp. 1313–1319, 1968, https://hal.science/hal-01537654.