| Author | Message | ||
     Benjamin Kaplan (kaplan) Junior Member Username: kaplan Post Number: 3 Registered: 08-2003 |
I have some questions concerning the "mathematics" involved by the flexPDEsolver. 1/- What are the differences between the "Finite difference method"(FDM) and "the finite element methods"(FEM = FlexPDE method ?) ? What are the relative advantages of these two methods especially when singularities are involved in boundaries conditions? 2/-I have heard about an other method the "Boundary element method"(BEM), what is the advantage and thedifference with the two others ? 3/-What are the principles of the integral calculations involved inFlexPDE? Is it as accurate as "Finite Difference Method" use to be? 4/- What is the principle of the discretization methods involved in the FlexPDE program? 5/-I don't really understand the concept of the "LANCZOS" parameter, could you explain this to me? 6/-The FlexPDE program seems to use the "Galerkin method" - What is the role of this treatment and it'sefficency towards other methods? Thank You if you can answer to these questions to understand the mathematical basis underneath the FlexPDE program and have a better use of this program.Thank you very much indeed. B. Kaplan. | ||
| Robert G. Nelson (rgnelson) Moderator Username: rgnelson Post Number: 38 Registered: 06-2003 |
There is a vast literature on the topics of finite difference and finite element methods, and I cannot replicate it in the space available here. I suggest you go to your library and find a book or two that address the subject in a way that appeals to you. In general, the finite element method is preferable over finite difference methods because a) the generation of discretizations can be done in a way that is applicable to complicated geometries and does not require gridding along coordinate directions, and b) the discretization is formed using integral relations like the divergence theorem which automatically guarantee flux conservation. FlexPDE uses a Galerkin finite element method, with quadratic and cubic basis functions. Petrov-Galerkin weighting is used in advection terms. Integrals in FlexPDE are formed by Gaussian quadrature rules of varying number of points, depending on the particular geometry and basis. The Lanczos method is one of a number of "bi-conjugate gradient" methods that are effective for non-symmetric matrices. |