Hi, all. I got a question on the symmetry of system matrices. I am looking for ways to solve Non-linear PDE using deal.ii and Dr. Bangerth's video lecture from 31.5~31.7 is greatly helpful to understand basic of this part.
In many elementary codes in tutorial lists, we have used Conjugate Gradient Method to solve the matrix. CG Method basically assumes that matrix A is symmetric. I also checked that deal.ii offers other solver method, GMRES solver for non-symmetric system matrix A. Question 1. My question is where will the non-symmetry arise? To be more specific, let's take a example of "linearized" non-linear minimal surface equation with Picard iteration (as Dr. Bangerth's lecture 31.65) <https://lh3.googleusercontent.com/-TYzXkQCHcTA/WAksZOt6tjI/AAAAAAAAA7E/kX0e1wXWcfMcKc-dDYeq2rOOmXi-YtHjwCLcB/s1600/Screen%2BShot%2B2016-10-20%2Bat%2B3.42.54%2BPM.png> where u is scalar function. Because we use previous solution u_k , the term "1/sqrt(1+(nabla_u)^2)" will work as a scalar coefficient that varies over the domain. Does this fact will make our system matrix non-symmetric? I have tried some mathematics to this.....and I think that this might be non-symmetric , but still cannot sure on this....... (and think that if it is non-symmetric, than we would better to use GMRES Solver instead of CG Solver) Question 2. Can we expect our numerical error will also behave ~O(h^p) for non-linear system that includes iterative method? Not only mesh spacing and shape function approximation degree, but also many other parameters might influence error. (e.g. mapping degree, and degree of gauss quadrature...) For Non-linear problem with iterative method, the tolerance between previous solution and solution might also affect error However, if I use enough small tolerance between abs(u_{k}-u_{k+1}), will the error will also show ~O(h^p)? Always thank you all and I am learning a lot form this amazing community!! Jaekwang Kim -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to dealii+unsubscr...@googlegroups.com. For more options, visit https://groups.google.com/d/optout.
