# comments on FDM, FEM, FVM, SM, SEM, DSEM, BEM

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 July 18, 2004, 06:35 comments on FDM, FEM, FVM, SM, SEM, DSEM, BEM #1 kenn Guest   Posts: n/a here are my comments on finite difference methods, finite element methods, finite volume methods, spectral methods, spectral element methods, discontinuous spectral element methods, and boundary element methods. there are two type of domain solvers, the direct and the indirect. the indirect approach includes boundary element methods. So far BEM only can be applied to simple equations like Helmholz or Stokes, but not Navier-Stokes, so it's not a general-purpose method. But I might have got a groundbreaking idea to apply BEM to any type of equations. I have 25% confidence. so, just be patient and wait for one or two years. Here let's talk about direct domain solvers, which includes, FDM, FEM, FVM, SE, SEM, and DSEM. FDM is a Taylor-polynomial method and uses exact fitting, and can be applied to arbitrary complex geometry by using global mapping of geometry, which is awkward. FEM can be in higher-order, and there are two types higher-order FDM, explicit in space (conventional) and implicit in space (compact). Higher-order FDM is clumsy, though the compact FDM relieves the problem a little bit. FDM can be locally conservative, but higher-order FDM will find extremely hard to be fully locally conservative. FEM is a Laglangian-polynomial best-fitting method, and also uses Lagrangian polynomial to have local geometric mapping, so that it's very nice for complex geometry. However, many other methods can also take this approach. It's very hard to make FEM, which means continuous FEM, locally conservative. But one may develop diconstinuous FEM to be locally conservative. However, since Lagrangian polynomials are not suitable for higher-order methods, a locally conservative discontinuous FEM makes little sense. FVM is the lonely truely locally conservative method, which can be regarded as an integral form of lower-order FDM. However, higher-order FVM is hard to realize. In FVM, there are two types approximation errors, during the evaluation of surface integrals and during interpolations. Many people try to improve the accuracy of these two processes. however, these efforts are likely to fail. because these efforts undermine the spirit of FVM, a simple and fully locally conservative method. all these above three methods are slow, and FEM is particularly slower. Because FEM is not locally conservative, and because FEM care about the whole domain (it use integration) while FDM and FVM care about selected points. In spectral method, you use complete & orthogonal polynomials, something Lagrangian polynomials never can compete. In SE, you may take a exact-fitting approach like in FDM and FVM, that is, you care some points, called collocation points. You may care about the whole domain and take a best fitting approach like in FEM, which is called Galerkin. Spectral method should be regarded as a h-type spectral element method, that is, it's a single-domain very-high-order spectral element method. However, people often call them separately. This is due to the history. So far, most spectral element methods take Galerkin approach, while spectral methods have two options. but, of course, secptral methods can take collocation methods. Discontinuous spectral element method is a locally conservative (but not completely conservative, to be explained) spectral element method. the best-fitting version is often called Discontinuous Galerkin, while the exact fitting version was called Collocation multidomain spectral methods. DSEM is simple the state of the art technique. and it will remain as the leading DIRECT domain solver for decades, possibly for ever. however, DSEM is not complete locally conservative. in a complete locally conservative method, the number of discrete unknowns should equal the number of discrete conservative equations. in DSEM, for one subdomain there are 3 discrete conservation equations for a 2-d incompressible flows, however, there are dozens of unknowns. the major advantage of SE, SEM, and DSEM over FEM and FVM is accuracy. the accuracy is gained during domain/surface integration and interpolation. and all these methods are more flexible than FDM. DSEM is a combinations of virtually all these direct domain methods. Godunov FDM and ENO-WENO FDM can find there images in DSEM; Lagrangian polynomails in FEM is used in DSEM for geometric mapping. the Galerkin in DSEM was originally used in FEM; the collocation in DSEM was originally used in FDM; the lowest-oder DSEM reduces to FVM; spectral is the foundation DSEM; Spectral element method is a continuous version or reduced version of DSEM. DSEM is very very flexible, can be used in arbitrary mesh, with some nodes of some element sitting between two nodes of another element --- most methods disallow to do that. numerical research in DSEM is still a big paper-generators. combination of DSEM with free-surface tracking techniques and flow-structure interaction is untouched. application of DSEM to multiphysics is widely new. conclusions: forget all other low-class direct methods, let's move on discontinuous spectral element method.

 July 18, 2004, 17:37 pros and cons of boundary element methods #2 kenn Guest   Posts: n/a two major cons: 1. some claim BEM reduces dimensionality by one consequently speed up numerical calculations significant. That's wrong! A direct numerical method such as FEM is considerably faster than BEM for 3-D internal flows. If we take N by N by N segments in each direction, the # of operation for BEM takes oder of N^4, while order of N^3 logN for FEM. that's because BEM produces dense matrix, while FEM produces sparse matrix. 2. So far, BEM works only for simple equations, because Green's function can't found for more complex equations. there is one pros: 1. BEM is particularly effective for external flows, such as potential flows and stokes flows. because you don't need cut a computational domain from the physical domain -- a common practice for direct domain methods. In stead, you just consider the surface of the flow domains, and you never need consider the surface at infinity.