Talk:Finite element method
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Discussion.
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Math tags
math should be typed using the math tag
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I agree.
Sturm-Liouville?
There is also something not clear about how the integration by parts is actually done. Something was left out there. (BTW does this have anything to do with Lu=g being of the Sturm-Liouville type?)
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No. When the operator is not Hermitian, there is still a bilinear form, except of course that it is not an inner product. One can obtain existence and uniqueness from the Lax-Milgram theorem, assuming that the bilinear form is coercive. If it is not, it is sometimes still possible to obtain a form of the Fredholm alternative.
When L is of the Sturm-Liouville type, the Dirichlet problem gives rise to a Hermitian operator and so the theory is nicer (and the linear solve is also easier -- conjugate gradient with a preconditionner can often be used.) If the boundary condition is not Dirichlet, the bilinear form is usually not symmetric, even if L is of Sturm-Liouville type. Loisel 11:46, 26 Jul 2004 (UTC)
Approximate
The benginning of this article states that FEM approximates solutions to PDEs. But isn't the result somtimes exact? For example; p-type elements with shape functions of order greater than two yield exact results of a beam simulation (and neglecting computer rounding errors). Its been many years since I had this class, so my memory might be fuzzy. Pud 00:46, 25 Jul 2004 (UTC)
I don't know what you're talking about, however it is possible in certain cooked-up examples for the numerical scheme to be exact modulo machine precision. However, this is true of almost every numerical method. Numerical differentiation, integration, ode solvers, pde solvers, root-finding, eigenvalue algorithms, singular value decomposition, etc... can all coincidentally be exact under certain circumstances. Loisel 11:42, 26 Jul 2004 (UTC)
Loisel, do you know what p-type elements and shape functions are? Pud 16:15, 26 Jul 2004 (UTC)
As I said, no, I don't. Loisel 11:44, 27 Jul 2004 (UTC)
- Maybe I have a dated vernacular. Anyway, P-type elements have polynomial shape functions that define the local stiffness matrix of the finite element. The size of the global stiffness matrix can be increased by refining the element mesh -or- by raising the order of the shape functions within the elements. So, consider simulating the beam differential equation; d2v/dx2 = M/EI, the closed form solution is a polynimoal. If the shape functions are a polynomial of at least the same order as the closed form solution then the finite element method will give exact solutions, I think.
- Mechanica and many other simulation softwares use p-type elements. This also allows refinement of local elements as needed without re-meshing. Pud 13:53, 27 Jul 2004 (UTC)
I know about piecewise polynomial basis functions on a triangular mesh, if that's what you're saying.
When solving an ODE like d^2v/dx^2=c, the solutions are v=cx^2/2+ax+b, for any a,b. Then one can cook up any number of numerical schemes to solve them exactly (that's what I was talking about in my first reply above.) For instance, the two-step method v[k+2]=2*v[k+1]-v[k]+c is exact in this example (if not entirely stable) even though in general it is not -- that is what I was talking about when I said "cooked up example." The FEM in this case can be written as an implicit method and without doubt some such schemes will be exact in this case. However, I'm fairly certain that the similar conclusion is false in the two variable case d^2v/dx^2+d^2v/dy^2=c because those functions are not polynomials. If c=0, one gets the harmonic functions, none of which are polynomials.
Erratum: of course some polynomials are harmonic.
Loisel 14:58, 27 Jul 2004 (UTC)
- Yes, piecewise polynomial basis functions on an element (that sum to one and describe the local stiffness matrix), though not specifically for triangular elements, are p-element shape functions in the commercial FEA software vernacular. This article should have a paragraph on p-elements and h-elements since they are the most common commercial method. I'll plan on doing this, after I've re-learned what I forgot fifteen years ago :) Pud 01:57, 29 Jul 2004 (UTC)
Terrible examples and explanation
It's awful! If I had something like that in university I would stop studying physics. Why not at least write the differential equation in it's native form first?
I hope the structure of the example, as well of the entire article, will be changed to something more comprehensive. (not signed)
- Well, this is an encyclopedia article, not a physics textbook. So, whoever wrote this article wanted to take it gently, as most of the Wikipedia audience don't have a good math background. Do you have more specific criticisms of this article? Oleg Alexandrov 01:27, 15 Jun 2005 (UTC)
- This explanation requires even more background knowledge than what we learn in university... User:Muxec
- I guess because the authors of this article wanted to write this from a math perspective. In physics, you guys don't worry about a lot of things mathematicians worry about. :) However, I would think this article could have been much more mathematical and much more technical than what it is now. Oleg Alexandrov 17:10, 18 Jun 2005 (UTC)