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.

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.

Sorry, I am A stupid chinese who speak stupid english. I found two many errors in my yesterday's post. The following is A modified version, in which some comments are expanded and more detailed.

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 types of domain solvers, the direct and the indirect. the indirect approach includes boundary element methods. In indirect methods, one significantly modifies the original systems mathematically, then discretize and solve it. Typically such a process involving GOING BACK from differential equations to integral equations and using the existing most powerful analytical technique for solving PDEs, the Green's function technique. 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. The other problem with BEM is the consequence of dense matrix. There are some efforts at Stanford, UIUC, and France to make the matrix sparse, such as fast multipole methods. I appreciate their effort, but doubt if they are heading in the right direction.

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. FDM 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, even 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. (to be explained later)

FEM is a Lagrangian-polynomial & best-fitting method, and also uses Lagrangian polynomial to make local geometric mapping, so that it's very nice for complex geometry. However, many other methods can also take this approach in dealing with complex geometry. One might comine the geometric mapping with FDM and create a new type of FDM for complex geometry. 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. Because, once you have improved FEM by discontinuous approach, why don't you put more time and make it higher-order? The true advantage of FEM is at its simplicity. Discontinuous FEM is neither simple nor very accurate.

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 violate 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 (by using element integration) while FDM and FVM care about selected points.

In spectral method, you use complete & orthogonal polynomials (such as Fourier, chebyshev, and Lengdre, but forget Hermitian), something Lagrangian polynomials never can compete. As order of polynomials increases, Lagrangian cluster in some regions, while spectrals distribute everywhere. So that, spectral can adjust its coefficient very fast to represent the solution, while Lagrangian need much and much more terms to represent solution accuractely. 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. The collocation approach is signicantly faster than Galerkin approach, if you use Lengdre polynomials; collocation is moderately faster than Galerkin if you use Fourier and Cheysheve. that's because in Galerkin spectral methods, the major calculation are performed in transform space, and those calculated numbers are not the solutions you want, but merely coefficients. So, you have to convert them into solutions, which is very time consuming for Legendre where fast fourier transform seems not possible. Note, converting from coefficient into solutions not just happens in the postprocessing stahe, you may have to do it during major processing. In Fourier and Chebyshev, you can use FFT to circumvent the issue. In Collocations methods, you consider everything in the computational space (non-tranform space). Or equivalently speaking, you care about solutions at selected points, but as you know, these numbers are always the same, no matter in the original space or the transform space.

Generally speaking, Galerkin is not as efficient as collocation. But, Galerkin seems more flexible ( I am not sure); and with lots of elements, you don't have very high order polynomilas on each element any more, and the problem with Lengdre Galerkin spectral method is no longer that severe. In Lagrangian Galerkin FEM, which cares about everything in the domain, the coeficients happen to be the solutions. So you don't have the problem associated with many other Galerkin methods.

Fourier polynomials works extremely well for periodic problems, but not well for problems with walls. overall speaking, Chebyshev is the best among the best for numerics.

Spectral method should be regarded as a p-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, spectral element 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 simply 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 completely locally conservative. in a completely locally conservative method, the number of discrete unknowns should exactly equal the number of discrete conservative equations. in DSEM, for one element there are 3 discrete conservation equations for a 2-d incompressible flows, however, there are dozens of unknowns. So, these unknowns satisfy 3 conservative equations and many weighted equations.

the major advantage of SE, SEM, and DSEM over FEM and FVM is accuracy & efficiency. 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 influence 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 for 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 vertices of some element sitting between two vertices of another element --- most methods disallow 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 turbulence and multiphysics is widely new.

conclusions: forget all other low-class direct methods, let's move onto discontinuous spectral element method.