Talk:Finite element method

From Wikipedia, the free encyclopedia

	Portal This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.
???	This article has not yet received a rating on the assessment scale. [FAQ]
???	This article has not yet received an importance rating within physics.
Help with this template Please rate this article, and then leave comments to explain the ratings and/or to identify its strengths and weaknesses.

1 History
2 Math tags
3 Sturm-Liouville?
4 Approximate
5 Terrible examples and explanation
6 Not much use to a non-mathamatician
7 Ben's rewrite proposal
8 Rewrite
9 Finite element analysis
10 spectral element method
11 Libraries
12 meshing
13 finite element method and finite volume method ?
14 triangles
15 equation (3)
16 CORRECTION NEEDED
- 16.1 An Equivalent Norm for $W_0^{m,p}(\Omega)$

[edit] History

How about John Argyris' contribution? I think there should be a relevant revision in the history section, provided that experts in the field also agree.

One short biography appears here: http://www.cimne.upc.es/webcimne/boletinECCOMAS/20040518.htm in word format. I do not know though if it can be made available to wikipedia...

http://www.mlahanas.de/Greeks/new/Argyris.htm

Dpser 17:57, 16 January 2007 (UTC)

--- Information (3 June 2008):

In my book

Kurrer, Karl-Eugen: The History of the Theory of Structures. From Arch Analysis to Computational Mechanics. Berlin: Ernst & Sohn 2008.

one can find the chapter " 'The computer shapes the theory' (Argyris): the historical roots of the finite element method and the development of computational mechanics" (pp. 619-672).

...And of course brief biographies of the pioneers of fem e.g. Argyris,...

Best regards, Karl-Eugen Kurrer —Preceding unsigned comment added by 212.202.96.83 (talk) 15:13, 3 June 2008 (UTC)

[edit] Math tags

math should be typed using the math tag

---

I agree.

[edit] 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?)

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)

[edit] 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)

[edit] 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 its 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)

Actually, I am a mathematician and active in the field of finite elements for about 15 years now. Therefore, I would not accept the disqualification above for me. Still, I must confess, I recognize the method only with difficulty from this article. A person trying to find out what it is will be left completely clueless. I can only recommend rewriting most of it, in particular

Change the Example to a Dirichlet BVP of the Laplacian.
Restrict to Hilbert spaces to keep the presentation simple.
Write the weak form by using test functions in $V$ .
What are those two discretization steps? Replace $V$ by $V h$ immediately.
There is no mass matrix involved in solving Poisson's equation. The right hand side vector can be computed directly. Actually, the computation of the coefficients $b k$ involves the inverse mass matrix.
There should be examples for the choice of the basis.

Guido Kanschat 13:42, 9 December 2005 (UTC)

These are all good points. I guess that periodic BCs were used because it allowed to make the connection to spectral methods. Definitely, do add an example giving a particular choice for the basis. You might also consider changing to 1d so that the PDE changes to an ODE, e.g.

u'' + u = 0

with boundary conditions

u (0) = u (1) = 0

; in this case, you can write out the final equation explicitly. I hope you have a go at it. -- Jitse Niesen (talk) 15:15, 9 December 2005 (UTC)

[edit] Not much use to a non-mathamatician

I was looking for a simple, lay-persons explanation of the difference between a finite element and a finite difference approach to solving a flow simulation equation. I certainly didn't get that. This is supposed to be an encyclopedia for everyone, how you can start an article with "assuming a knowledge of calculus" is beyond me - what percent of the population actually has a working knowledge of calculus? Your approach is arrogant and exclusive, hopefully someone will find a more approachable way to describe these points soon, if Hawkins can describe the big bang then you should be able to describe this.

JohnH

And you are complaining to whom? Be happy there was a kind soul who wrote this article according to what he/she knew. I am a mathematician, and I find this article very acceptable. You don't like it, so please change it. But be advised however that this is a well-written article (even if maybe a bit higher level than what you wish). So if you feel you can do a good job, be bold! Otherwise let us wait, as you say. :) Oleg Alexandrov 15:34, 21 July 2005 (UTC)

It would be helpful to state what kind of information you seem to be lacking, and perhaps a brief explanation of your background (I might be wrong, but this isn't a subject typically even visited by a person with no afilliation to natural sciences, I suspect perhaps you might be an engineer?). As an encyclopedia article it is supposed to be brief and can not treat everything. However personally I think the subject of FEM should be treated as a subset of WRM and build upon such. Basically as an applied matemathician I would like it to treat the mathematics more thoroughly. Basically I think you come off a bit rude, and you would be well advised in adressing such issues a bit more humbly. That aside, I think a comparisson with FDM would be a good addition to this article, and I will make a short admendment about this imediately. Bfg 02:03:27, 2005-08-19 (UTC)

Oleg Alexandrov, I saw you made an extra indentation to the above comment, I would say that it's normal to reply with one more indent than the person you are replying to. Are you sure this is correct indentation style? To clarify, the above statement is directed at JohnH. Anyway, I added the above mentioned section, it is brief and perhaps not very exhaustive. Please adjust it as you might see fit. Bfg 03:22:18, 2005-08-19 (UTC)

OK, I am not sure at all. I put it back. Oleg Alexandrov 03:25, 19 August 2005 (UTC)

Bfg, tusen takk for your addition. I took the liberty of also mentioning an advantage of FDM, otherwise people will wonder why it is used at all ;) By the way, what do you mean that it's easier to achieve higher order in FEM? I think that's also pretty easy in FDM. Are you referring to stability problems, or performance issues because the stencil becomes too big? Perhaps the comparison between different methods would fit better in numerical partial differential equations.

I do think the article is not very good though, because it only describes one example and does not say how to generalize to other elements or equations, nor give a definition of FEM (of course, it is not easy to formulate a definition with which everybody agrees). So, Bfg, if you want to change the article you have my blessing. -- Jitse Niesen (talk) 13:17, 19 August 2005 (UTC)

Well, you're welcome, I'm sorry I don't know how to say you're welcome in Dutch though. I fully agree about the ease of implementation for FDM. As for the higher order elements, the quick answer is I haven't really thought it properly through. (There's probably something that could be said about being to bold in the middle of the night.) When forced to think about it I realize that I can justify my position with two reasons. The first is redundant and goes back to the treatment of complex geometries. The second is the easiness of generating p-type elements of higher order. I think I could explain the procedure with relative ease for a good student in junior high, using only elementary algebra and the bisection method (That's the generation of basis functions, not the entire FEM). Now in FDM, using higher order aproximations of equations would require some basic understanding of analysis.

I'm not going to remove my statement, but feel free to do so if you think my justification is too weak. I've taken a look at your backgrounds and I humbly defer this decision to higher authorities ;) . Bfg 21:20:05, 2005-08-22 (UTC)

I thought a bit about it and I decided to delete that statement. To create high order in FEM, you take a basis with polynomials of a high enough degree and you calculate the mass and stiffness matrices. In FDM, you have to find the stencil, which you can do by requiring that the stencil is exact for polynomials of high enough degree; the procedure is the same in one and multiple dimensions. I'd say it's about the same in both cases.

Don't defer to my authority though. I have no experience with FEM and I had to look up what WRM is. And "you're welcome" is "graag gedaan" in Dutch. -- Jitse Niesen (talk) 18:31, 1 September 2005 (UTC)

[edit] Ben's rewrite proposal

I agree that this assumes a lot of math background. It's great to have a rigorous explanation, but a layman's explanation and an engineering explanation would be good. I propose this article get forked so that this page, Finite element method is a layman's introduction to the history and uses of the technique, then provide links to Mathematical treatment of the Finite Element Method and Engineering treatment of the Finite Element Method much like Tensor has Classical treatment of tensors, Tensor (intrinsic definition), and Intermediate treatment of tensors. —BenFrantzDale 03:40, 22 November 2005 (UTC)

An explanation along the lines of Engineering treatment of the Finite Element Method would be excellent and I hope you will find time to fill in the gaps. I never thought this through properly, but I can imagine that this gives exactly the same method as the mathematical treatment.

However, I am not so sure about forking the article. If the combination history/background + engineering treatment + mathematical treatment gets too large to fit in one article, then I'd fork off the maths and leave the engineering bits in, because the engineering treatment will probably be understood by far more people.

By the way, what do you think about finite element analysis? In my experience, "finite element method" and "finite element analysis" mean pretty much the same, so it seems strange to have a separate page on it. -- Jitse Niesen (talk) 13:46, 22 November 2005 (UTC)

You may be right that forking the way I suggested could get ugly. Still, I think it is important to maintain the rigerous mathematical treatment even as we create (hopefully) a good engineering description. (I find it notable that here at RPI, there are three introductory graduate-level finite element classes—one in Mechanical Engineering, one in Math, and one in the Computer Science department.)

I think finite element analysis is a great start for a laymans explanation. If someone wants to know “what does a Finite Element package do?” that's the article they should see. I'm not an expert, but I don't have any mental distinction between FEA and FEM, aside from them being slightly different parts of speech.

I'll modify the proposal as follows:

The bulk of this page should move to Mathematical treatment of the finite element method
Finite Element analysis should be moved here
Engineering treatment of the Finite Element Method should be fleshed out and appended to the layman's description on this page.

That way this page would read as a layman's introduction followed by an explanation that assumes linear algebra, Newtonian mechanics, and calculus. (This explantion may brush against the calculus of variations but not assume knowledge or go in depth. Does this sound like a reasonable direction? (I'm curious what Oleg Alexandrov thinks, in particular, since he seems to be a major contributor to the mathematics portion.) —Ben

I contributed almost nothing to this article, all I remember is that I put the picture. Your suggestion sounds reasonable, but I would like to say that it would take you a lot of work, and I would suggest you first move the existing text to Mathematical treatment of the finite element method before you start working on the new article. That is to say, I would be interested to first see how you are progressing on the more elementary article before you attempt to rewrite the existing text which does not look too bad in my opinion.

And I agree with Jitse that it does not make much sense to have finite element method and finite element analysis separate. Oleg Alexandrov (talk) 21:43, 22 November 2005 (UTC)

Guys, ... please, hear a humble oppinion from an electrical engineer. I just started looking into FEA recently to simulate electrical field problems and needed good info on FEM. This Wikipedia article was by far best what I could find on the Web after a few days of Google search. But... I could not disagree that the treatment here of FEM is more "mathematically abstracted" than it needs to be for the "popular engineering audience" (although one can argue the FEM/FEA audience is popular in the sense of guys going out on a Friday night... :) Anyway, I applause the effort for more "engineeringly" oriented article. And I hope to be able to contribute to this if I can. Three specific comments:

a) In my humble oppinion there is a difference between FEA and FEM. It is obvious to even an "absolute beginner" (as myself) - the "Analysis" includes much more stuff than the "Method" of solving a PDE (which is FEM), namely: choice of a physical model, problem formulation (related to this choice), applying skills to define conditions/analyse the results of the simulation in terms of a particular application (physics, mechanics, electrical engineering etc.) As I see it: Although FEA is using FEM for solving the equations at hand - FEM remains "simply" a way to solve PDE's etc. All the rest - choice of basis, test functions, proper approximation etc. should be contributed to the "Analysis". That may sound superficial, but is what I was able to figure out so far. Just my way of understanding it.

b) The article IS better to be sub-divided (not same as split) in Engineering treatment of the Finite Element Method (or, better said "Engineering applications of FEM") and Mathematical treatment of FEM. As I already commented, I view FEM as "simply" a good, generic method for solving PDE's (useful not just because it is a clever mathematician's idea (Richard Courant, 1943) but because it IS a very useful computer tool). So, I agree, that FEM can be treated "more mathematically extensively" and described as such. But then, a page on "FEA for engineers" needs to address all these other "application details" that I already mentioned make up FEA in addition of appying FEM to solve a PDE of a model.

A good example of how FEM is applied for an engineering FEA will help here as suggested by others, not just to keep non-mathematical audience's attention but simply because FEA/FEM strenght is in fact in the ENGINEERING APPLICATION of the method, really. Do you agree?

c) The current FEM page does mention the choice of basis and suggests the triangular tessalation (which I see a very common-sense choice so far in practice) but then, it really stops there. It is missing a VITAL piece in my oppinion - to answer "what's next?". For instance, the "Algorithm" stops at a pretty vague p.5 ("Possibly convert the vector a back into a solution u (e.g. for viewing with a graphical computer program"), and misses to mention anything about how the boundary conditions are applied or how the test functions are approximated to actually get back to u(x,y). Anyone trying to apply this to his/her engineering task would be interested in this a lot!

Despite all, this, still, was the best info I could find off the Web quickly and understand what FEM is about - big thanks to the aurhor of the article, too bad Wikipedia does not show author's (including contributing authors) info and contact(s)! --Momchil 22:59, 22 December 2005 (UTC)

P.S. The draft (Engineering treatment of the Finite Element Method) I mentioned above lost me completely, I feel I liked the mathematician's view on FEM better. This stub/draft starts very well, has good partitioning but the energy analogy feels a bit too much. If the example is to be from mechanics (elastostatics) - fine, but please keep it in this domain (although the analogy may be really good one). The thing is that if someone a kind of familiar with static problems in mechanics is trying to figure it out and is about to understand it, this analogy can throw him off, you know. Please complete "A single element" and "Combining elements" as these are vital to the understanding of all. "Notation" section is really good to have, please complete. —the preceding unsigned comment is by Momchi (talk • contribs)

---

I'm the mathematician who originally wrote much of this article. I have also read Engineering treatment of the Finite Element Method and I consider it gobbledygook. If further examples are desired, then further examples should be added. If a nontechnical section is desired, then a nontechnical section should be added. However, if you use the word "elastostatics", you can rest assured that you are in fact talking about your own specialized application of the FEM which is simply more complex and less insightful than the current article.

Just to be clear: I vehemently oppose Ben's proposal of rewriting the FEM article in the style of Engineering treatment of the Finite Element Method.

Loisel 23:02, 26 December 2005 (UTC)

I also so far like the existing article here much more than what is at the engineering treatment of the Finite Element Method article. First, that one looks way way too long for an encyclopedic article, second the writer of that article did not get to the point yet, after pages and pages written. I would prefer more if this article stays the way it is, and if somebody would implement Guido's suggestions, see #Terrible examples and explanation above. Oleg Alexandrov (talk) 03:18, 27 December 2005 (UTC)

[edit] Rewrite

Is this better?

Loisel 19:04, 27 December 2005 (UTC)

I read all the article, and it explains things very well. Thank you!

One question. The link to torsion is ambiguous. Would you consider chaning it to something more specific?

There needs to be a picture of the basis elements in 1D, I hope to do it sometime. Oleg Alexandrov (talk) 21:36, 27 December 2005 (UTC)

I also think a picture of the 1D elements would be nice. Loisel 01:03, 28 December 2005 (UTC)

It's on the top of my to do list! :) Oleg Alexandrov (talk) 01:40, 28 December 2005 (UTC)

[edit] Finite element analysis

How to fit Finite element analysis in this article? --Abdull 16:30, 21 February 2006 (UTC)

Finite element analysis involves the application of finite element methods to specific problems. Temur 23:40, 9 October 2007 (UTC)

I don't think Finite Element should automatically redirect to Finite element analysis. It made me miss this page for a long time. —Preceding unsigned comment added by Knepley (talk • contribs) 22:11, 30 January 2008 (UTC)

[edit] spectral element method

I think there should be a seperate article on the spectral/hp element method, which is NOT the same as the current spectral method article, as this FEM article only mentions SEM in passing and does not give ANY details

there is also no mention of testing functions, galerkin method, etc. also no references to fem textbooks. i will add those in when i get a chance --anon

There is a Galerkin method article. I suggest you write more details here before making big changes to the article. Oleg Alexandrov (talk) 02:43, 30 June 2006 (UTC)

The definition of the 'spectral element method' given in the article is incorrect. The spectral element method uses a frequency domain formulation of the stiffness matrix ('dynamic stiffness') and was developed for use in dynamics problems. (e.g. wave propagation in structures). Nowhere in the literature is the mere use of higher degree polynomial basis functions considered a 'spectral' method.

Those are called 'higher order finite elements'. Temur 23:38, 9 October 2007 (UTC)

[edit] Libraries

How about some links to free libraries (from the polish wikipedia)

Diffpack
Z88
SLFFEA
YADE
FEniCS
deal.II
getFEM
libMesh
freeFEM
Code-Aster
Impact
IMTEK Mathematica Supplement (IMS)
Calculix
Elmer
OOFEM -- a darmowy, wolny, obiektowy pakiet MES ogólnego zastosowania
IFER

[edit] meshing

Does the subject of finite element meshing deserve its own article? I'm thinking so. It's distinct enough from the computer graphics description. Ojcit 19:21, 2 October 2006 (UTC)

[edit] finite element method and finite volume method ?

Is the finite element method somehow related to the finite volume method ? —The preceding unsigned comment was added by Domitori (talk • contribs) 01:11, 17 December 2006 (UTC).

They are both method of numerically approximating the solution to differential equations. I'm not sure how the formulations might be similar beyond that. - EndingPop 15:12, 17 December 2006 (UTC)

They both can be fit into Petrov-Galerkin approximation framework. You can think of Finite Volume Method as a FEM with special type of trial function spaces. Temur 23:37, 9 October 2007 (UTC)

[edit] triangles

The section on how to choose the basis is in my opinion too much focused on triangles, since the FEM can be applied to general elements, for example also to quads etc.

I would agree. It needs to be much more general. - EndingPop 17:11, 2 May 2007 (UTC)

[edit] equation (3)

is referenced in the text, but I can't find it (should be somewhere between (2) and (4) I guess). --147.122.2.207 10:51, 26 January 2007 (UTC)

Never mind, I think I found it. I have restored the tag. --147.122.2.207 10:54, 26 January 2007 (UTC)

Speaking of references: I liked the article so far but there is one thing I couldn't help noticing...There are virtually NO references (only one link to cover the history part in general). Not that I doubt the accuracy of what was written but references are still important. Could contributors if they have any free time look into referencing the bits they wrote? I for one would find it useful to direct my wider reading. Cheers, Pl4t0 02:43, 12 May 2007 (UTC)

Behshour 13:56, 27 June 2007 (UTC)

[edit] CORRECTION NEEDED

1. The bilinear form does not define an inner product in $H_0^1$ since it does not satisfy the "homogeneity" property of the inner product: For any inner product $< .,. >$ we must have: $< u, u > = 0$ if and only if $u = 0$ , but this is not the case here since $< u, u > = 0$ only implies that $u' = 0$ . Consider the following $L 2(0,1)$ function: $u = 3$ on THE OPEN INTERVAL (0,1) and $u (0) = u (1) = 0$ . Obviously this is an $L 2(0,1)$ function whose $L 2(0,1)$ norm equals 3. Its derivative $u'$ is identically zero with a zero $L 2(0,1)$ norm. Therefore $u$ is in the space $H_0^1(0,1)$ . At the same time, $B (u, u) = 0$ , but $u$ is not identically zero. Although this bilinear form does not generate an induced norm, but it generates a semi-norm, and it may be considered a "Minkowski functional" instead of an inner product. It's however noteworthy that another bilinear form on the space $H_0^1$ , namely, $\int_0^1 u'v'+uv$ does define an inner product and its induced norm is the standard Sobolev space norm. In this particular argument, however, whether the original bilinear form defines an inner product or not is irrelevant. If you are worried about Reisz representation, all it has to be is a bounded linear functional. Therefore I suggest that that remark be omitted.

2. In reference to the space $H_0^1(0,1)$ , it is possible that in DIMENSION 1, and due to embedding theorems of Sobolev, along with the reflexivity of the Hilbert space, this space (or its closure) may be associated with the space of functions of bounded variation. But when one is defining a space, the definition must be as fundamental as possible. Associations do not replace definitions and they come next. The Sobolev space $H_0^1$ is different from the space of functions of "bounded variations" or " absolutely continuous" as it has been changed back and forth; otherwisw, it would be named that way. $H_0^1(0,1)$ must be defined as: $H_0^1(0,1)={u:u,u'\in L^2(0,1); u(0)=u(1)=0}$ .

Please note that it's important to verify the above as soon as possible, and to make corrections as needed since otherwise, it would be misinforming.All Best.Behshour 14:17, 27 June 2007 (UTC)Behshour 14:23, 27 June 2007 (UTC)

---

$\int \nabla u \cdot \nabla v$ is equivalent to $\int \nabla u \cdot \nabla v + uv$ , I refer the reader to "Sobolev Spaces" by Robert A. Adams and John J.F. Fournier for details. In my copy, one find the following on page 183.

[edit] An Equivalent Norm for $W_0^{m,p}(\Omega)$

6.29 (Domains of Finite Width) Consider the problem of determining for what domains $Ω$ in $\mathbb{R}^n$ is the seminorm

$|u|_{m,p,\Omega} = \left( \sum_{|\alpha|=m} ||D^\alpha u||^p_{0,p,\Omega} \right) ^{1/p}$

actually a norm on $W_0^{m,p}(\Omega)$ equivalent to the standard norm [...]

We can easily show the equivalence of the above seminorm and norm for a domain of finite width, that is, a domain in $\mathbb{R}^n$ that lies between two parallel planes of dimension (n-1). In particular, this is true for any bounded domain.

6.30 THEOREM (Poincaré's Inequality) If domain $\Omega \subset \mathbb{R}^n$ has finite width, then there exists a constant $K = K (p)$ such that for all $\phi \in C_0^\infty(\Omega)$

$||\phi||_{0,p,\Omega} \leq K |\phi|_{1,p,\Omega}.$

[...]

6.31 COROLLARY If $Ω$ has finite width, $|\cdot|_{m,p,\Omega}$ is a norm on $W_0^{m,p}(\Omega)$ equivalent to the standard norm $||\cdot||_{m,p,\Omega}.$

---

That said, the reason why I skipped this explanation in the text is that it would considerably lengthen the discussion and this is not really about finite elements, but rather, this is a fact about Sobolev spaces or Elliptic boundary value problems.

Loisel 18:22, 27 June 2007 (UTC)

Poincaré's inequality is probably worth adding to Wikipedia. Loisel 18:26, 27 June 2007 (UTC)

Haha! It's already there. Loisel 18:27, 27 June 2007 (UTC)

The actual theorem I have above is also known as Friedrichs' inequality. Loisel 18:43, 27 June 2007 (UTC)

I just want to add that the counterexample 1 does not work since the function is not in $H_0^1(0,1)$ . Temur 23:34, 9 October 2007 (UTC)

Categories: Unassessed physics articles