# Talk:Finite element method

## Disappointingly poor quality article

This is a fine example of an article that represents the worst of all worlds - it fails to accurately present either the mathematical, physical or engineering aspects of FEM. It is overly simplified for applied mathematicians, incomprehensible for lay persons and useless for the practicing engineer/analyst. There are numerous blatantly incorrect / misleading statements and glaring errors of omission, and the level of presentation alternates between sophomoric and overly pedantic. Students and laymen would be well advised not to attempt to learn anything from this article!

I find myself in complete agreement with the two above statements. I humbly suggest that this page be split up into multiple independent pages on the subject from the differing perspectives that would make use of this subject matter (mathematicians, physicists, engineers...) so that all three of those groups could have a page replete with useful information without the current requisite diversions into other fields' terminology and conventions. Put simply, this topic is far too large for one page fully describe the scope of. Bobxii (talk) 17:50, 9 November 2010 (UTC)

I recently edited the article (section 2.1) by removing what was there with a few paragraphs written by Barna Szabo and Mark Ainsworth. It has been reviewed by Ivo Babuska, Ernst Rank and Alexander Duester among others considered the best in the field. Whomever rejected the edit should go back and learn, not only about the finite element method, but also about humility. He dismissed by stating that "a clear technical description works better than theoretical opacity", for a text written by 5 of the best in the field...if it is opaque it may be because your lenses are dirty dear Woodstone... I've seen some of the responses to the criticism be based on how much they are supposed to know instead of recognizing that they are other that may know even more and check their facts. It is truly disappointing to see that people egocentrism is used as a reference. I'll try to set the edit back...if unsuccessful I'll simply create another alternative page and we will let people decide which one contains valuable information. — Preceding unsigned comment added by Snervi (talkcontribs) 17:05, 27 November 2014 (UTC)

I don't see why the paragraphs you say were written by Barna Szabo and Mark Ainsworth can't be added rather than have to replace what was there. They seem complementary to me.--agr (talk) 02:27, 28 November 2014 (UTC)
Unfortunately, the previous text could not be edited because it started from the wrong premise. There were misleading statements on the nature of the Finite Element Method that come from constructing the idea of the method (and terminology) from the element up, when it should be the other way around, arriving to the element formulation after formulating a variational problem and a discretization strategy. The element based approach, used by early investigators, is now considered obsolete. It has hindered the understanding of what a finite element method is. — Preceding unsigned comment added by Snervi (talkcontribs) 22:39, 29 November 2014 (UTC)
As this reference makes clear, both approaches are still taught and used: http://www.colorado.edu/engineering/cas/courses.d/IFEM.d/IFEM.Ch01.d/IFEM.Ch01.pdf Wikipedia tries to present different approaches to a subject as long as they are supported by reliable sources, not pick the best one. If you have published reliable sources for your claims of the superiority of the variational viewpoint, the article can be structured to say something like, "Historically FEM was viewed as blah blah blah,. A more modern approach is blah blah blah, which as the following benefits..."--agr (talk) 23:26, 29 November 2014 (UTC)
While it is understandable that it could be seen that way, there are no separate approaches and there is no superiority of the variational approach as it is one and the same. It is the way chosen to present the topic that can lead to misuse and misinterpretation. I agree with your statement from the historical perspective and acknowledge that "The history of the method" section should describe how the method was originally implemented, the shortcomings of that original implementation and how it was given a full mathematical framework later on. The idea to pick a superior formulation is flawed since (as I said before) there is just one, however, from the point of view of the use and implementation of the method, there are several ways to get it wrong. — Preceding unsigned comment added by Snervi (talkcontribs) 00:29, 30 November 2014 (UTC)
I still do not see why your overall description and the previous version cannot coexist. You need to provide published sources for any critique of what is in the original section. Simply asserting the superiority of your version is not enough. And continuing to revert other editors work is considered edit warring and can result in your be blocked for editing.--agr (talk) 05:22, 30 November 2014 (UTC)

## 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)

I think that the extremely significant role of O. C. Zienkiewicz has been overlooked here. He is generally considered a key pioneer, and, as rightly stated on his Wiki page: "His books on the Finite Element Method were the first to present the subject and to this day remain the standard reference texts."

Kurvenbau (talk) 08:42, 13 May 2009 (UTC)

I think that the works in matrix analisys of structures from Mohr and Maxwell in the XIX century must be taken in account. This pioneers works can help to the reader to understand the basis of the method because it was basically FEM with 1D elements. --Cometo22 (talk) 08:43, 8 March 2011 (UTC)

## Math tags

math should be typed using the math tag

---

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?)

--

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 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

1. Change the Example to a Dirichlet BVP of the Laplacian.
2. Restrict to Hilbert spaces to keep the presentation simple.
3. Write the weak form by using test functions in ${\displaystyle V}$.
4. What are those two discretization steps? Replace ${\displaystyle V}$ by ${\displaystyle V_{h}}$ immediately.
5. 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 ${\displaystyle b_{k}}$ involves the inverse mass matrix.
6. 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. ${\displaystyle u''+u=0}$ with boundary conditions ${\displaystyle u(0)=u(1)=0}$; 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)
Five years later, and frankly, this article is still terrible. Guido's remark that one can only recognize the method at all from this article with great difficulty is spot-on. There are plenty of very intelligent people discussing the most banal of things on this discussion page - the "underlooked role" of one person or another in its history, etc. - can't we improve this article so that it is worth something? The article is terrible, folks, absolutely terrible. If making little changes is not going to improve the situation, then please, someone be bold and rewrite the whole thing. -24.13.162.248 (talk) 00:59, 23 September 2010 (UTC)
What are you waiting for? Go ahead, make improvements if you have the ability (but please spare us with your opinion on how incapable all other editors are)! Tomeasy T C 06:55, 23 September 2010 (UTC)

## Not much use to a non-mathematician

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)
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)

## 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:
1. The bulk of this page should move to Mathematical treatment of the finite element method
2. Finite Element analysis should be moved here
3. 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)

## 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)

## 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 (talkcontribs) 22:11, 30 January 2008 (UTC)

## 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)

## 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)

## 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 (talkcontribs) 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)

## 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)

## 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)

## CORRECTION NEEDED

1. The bilinear form does not define an inner product in ${\displaystyle H_{0}^{1}}$ since it does not satisfy the "homogeneity" property of the inner product: For any inner product ${\displaystyle <.,.>}$ we must have: ${\displaystyle =0}$ if and only if ${\displaystyle u=0}$, but this is not the case here since ${\displaystyle =0}$ only implies that ${\displaystyle u'=0}$. Consider the following ${\displaystyle L2(0,1)}$ function: ${\displaystyle u=3}$ on THE OPEN INTERVAL (0,1) and ${\displaystyle u(0)=u(1)=0}$. Obviously this is an ${\displaystyle L2(0,1)}$ function whose ${\displaystyle L2(0,1)}$ norm equals 3. Its derivative ${\displaystyle u'}$ is identically zero with a zero ${\displaystyle L2(0,1)}$ norm. Therefore ${\displaystyle u}$ is in the space ${\displaystyle H_{0}^{1}(0,1)}$. At the same time, ${\displaystyle B(u,u)=0}$, but ${\displaystyle u}$ 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 ${\displaystyle H_{0}^{1}}$, namely, ${\displaystyle \int _{0}^{1}u'v'+uv}$ 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 ${\displaystyle H_{0}^{1}(0,1)}$, 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 ${\displaystyle H_{0}^{1}}$ 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. ${\displaystyle H_{0}^{1}(0,1)}$ must be defined as: ${\displaystyle H_{0}^{1}(0,1)={u:u,u'\in L^{2}(0,1);u(0)=u(1)=0}}$.

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)

---

${\displaystyle \int \nabla u\cdot \nabla v}$ is equivalent to ${\displaystyle \int \nabla u\cdot \nabla v+uv}$, 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.

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

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

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

actually a norm on ${\displaystyle W_{0}^{m,p}(\Omega )}$ 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 ${\displaystyle \mathbb {R} ^{n}}$ 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 ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ has finite width, then there exists a constant ${\displaystyle K=K(p)}$ such that for all ${\displaystyle \phi \in C_{0}^{\infty }(\Omega )}$

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

[...]

6.31 COROLLARY If ${\displaystyle \Omega }$ has finite width, ${\displaystyle |\cdot |_{m,p,\Omega }}$ is a norm on ${\displaystyle W_{0}^{m,p}(\Omega )}$ equivalent to the standard norm ${\displaystyle ||\cdot ||_{m,p,\Omega }.}$

---

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 ${\displaystyle H_{0}^{1}(0,1)}$. Temur 23:34, 9 October 2007 (UTC)

## Examples of FEM Codes

So how many of these codes are we going to add to the page? Is there WP policy regarding this sort of thing, because I feel like this is getting out of hand. - EndingPop (talk) 16:38, 18 June 2008 (UTC)

## The merge

Recently finite element analysis was merged with finite element method to create a new finite element methods. I think the merger was a good idea, but I don't understand why the new article was created with plural. Everywhere I've seen the method is written in singular, even if of course there are many variations of the method.

To the author of the merge: can you please explain how you merged things? I tried to figure out what you did from the diff, but all I see is a sea of red. Did you remove significant information?

Thanks. Oleg Alexandrov (talk) 03:25, 3 July 2008 (UTC)

I suppose you might claim there is only one Finite element method, but in actuality that isn't really true. I would suggest looking at the following books Brenner-Scott, Brezzi-Fortin, Girault-Raviart books or I even see books with the title "The finite element method" and then immediately talk about different finite element methods. Finite element methods are really about putting many pieces together. For example just using Ciarlet's definition gives many possible methods for making an element (which by the way is not just a subregion of the mesh). If you go into the literature a bit more you will find people that assemble the elements in completely different ways, the standard is based on the minimization in an L2 norm but I've seen L1 norm, and least squares minimizations as well. The previous articles basically presented the material as if first order lagrange polynomials were "The Finite Element Method" but in actuality this is a misrepresentation of the field, but perhaps I should bow to convention.

The merge which I proposed here, I wanted to make the material for finite element methods much cleaner. The finite element anaylsis article really was more about computer-aided engineering or scientific simulation and current trends in engineering. The finite element method article presented a mathematical view that, while true for a specific case, did not represent the general framework given, not to mention I found several errors along the way. IMHO the article should give a view of what the methods are, a bit about where they are used and why, and maybe some mathematics. I tried to combine these two pages to do that, unfortunately it might cause some controversy about what is and isn't acceptable. I would propose another page or maybe a series of pages on the different technical parts of the process, perhaps a page about finite element assembly, another on mesh generation and so forth.

-Art187 (talk) 09:25, 3 July 2008 (UTC)

Also to conform to standard you need to put a link to the old finite element analysis talk page here. Something like:

Article merged from Finite element analysis: See old talk-page here -Art187 (talk) 09:55, 3 July 2008 (UTC)

Also why is this a physics article? It has as much to do with physics as multiplication. -Art187 (talk) 09:58, 3 July 2008 (UTC)

Hello, I wrote much of the original article. I think it should be called "Finite Element Method". The Physics tag is probably because a lot of physicists are interested in this. I was the main editor, and I'm a mathematician, but I didn't tag it. Loisel (talk) 19:23, 3 July 2008 (UTC)

Of course there are different elements leading to different methods. But that doesn't mean the title of the article should be plural. We also have real number and Hilbert space (in the singular), because one of our naming conventions is that we prefer singular nouns. Obviously, the article should not give the impression that all methods use continuous piecewise linear approximants on a triangular grid.
Is it really true that finite elements are more general than finite differences? I can't find such a bold statement in Brenner and Scott, the reference given. I know that the simplest finite difference methods can also be considered as finite element methods, but I think that this is not true for all finite difference methods, especially when taking boundary conditions into account. -- Jitse Niesen (talk) 19:53, 3 July 2008 (UTC)
FD isn't a generalization of FEM. I was looking at the same thing yesterday, but I was too lazy to delete it. It's now deleted. What is true is that if you use an orthogonal lattice, the stiffness matrices are the same, but the mass matrices are not. If the lattice is not orthogonal, the proper generalization of FD is finite volume method, which is not based on the same idea as the FEM. Loisel (talk) 10:54, 4 July 2008 (UTC)
Point well taken Jitse, but perhaps moving from "the" Finite element method to "a" finite element method is more appropriate. -Art187 (talk) 20:33, 3 July 2008 (UTC)
I'm not sure I understand how boundary conditions cannot be modeled the same with FEM as FDM, as far as the different schemes you can make different weighted polynomial spaces that implement most of the FDM that I have seen, sorry no reference for this. Furthermore Courant's paper derives FEM by applying Ritz-Rayleigh to FDM, then there is a section in Brenner-Scott that derives FDM from FEM, (sorry I don't have the book anymore to give a page number). Furthermore FEM can handle more reference shapes and polynomial spaces. These are the reasons I added the statement about FEM being more general. One thing that I took out of the article (or tried to) was this notion that FEM handles more domains that FDM and FDM has no possibility for h-adaptivity. I also don't like the statement about FDM discretizing the operator versus the solution, these two things seem to be equivalent to me, if you discretize the operator with 1st order polynomial then it is equivalent to using linear polynomials as your solution space. I wanted to add some things on finite volume methods (which also were created as a finite element method) but I don't have the references off hand. -Art187 (talk) 20:33, 3 July 2008 (UTC)
I have no particular preference for FEA vs FEM as an article title, but I really do have a bit of a problem with FEMs, since that is not a natural article title for people who are looking for general information. So, if you insist that FEMs is prefered to FEM, then I should rather the article was called FEA and have done with it. To be honest I find FEMs to be a bit silly, any number of analytical approaches involve breaking things up into bits and then sticking them back together. Greg Locock (talk) 01:34, 4 July 2008 (UTC)
Greg, I believe the debate is over for FEMs versus FEM, I do not follow your reasoning for FEA though. You are correct that most numerical methods break things into parts and put them back together again. I posit that this is not what makes a method a finite element method, and is one of the major misconceptions laid down by the articles on FEM at wikipedia, instead look at Ciarlet's definition for which I gave in the article. -Art187 (talk) 07:01, 4 July 2008 (UTC)
On the difference of finite differences and finite elements: The only thing I can find in the book by Brenner and Scott is that FEM with continuous piecewise linear elements and FDM with second-order central differences coincide for the ODE u'' = f if a uniform grid is used (Section 0.5). -- Jitse Niesen (talk) 13:48, 4 July 2008 (UTC)
I also believe the finite difference method is not just a particular case of the finite element method. Many problems can't be written in a variational formulation (e.g., high order non-linear non-elliptic equations), and then one can't talk about finite element. Also, there are things as upwind schemes, various discretization tricks, which can't be expressed with finite elements. So, finite element and finite difference are distinct methods from what I know, although for simple problems both approaches can yield the same discrete system of equations. Oleg Alexandrov (talk) 17:04, 4 July 2008 (UTC)
Why can't you use FEM for high order, non-linear, and/or non-elliptic equations? Now there are definitely differences in the linear solvers and error bounds, but this exists in FEM as well. I solve non-linear equations with FEM all the time (in fact Girault-Renault is a whole book about FEM with a non-linear equation) and the second hit on google brings up a paper that does upwind methods for hyperbolic equations. Of course here SUPG is looking at the residual for the upwind knowledge, a far more robust method than the weighted quadrature that CFD uses, but one can do weighted quadrature with FEM too. For higher order see Ŝolín book (I haven't read it I just saw 4th order on the table of contents). The point here is not to say that FDM is always arrived at by FEM method of analysis but the claim that FEM is more general is that it can do everything FDM can do plus. That plus includes things like use more reference shapes and larger number of function spaces. -Art187 (talk) 19:44, 5 July 2008 (UTC)
Finite element can't work on problems which can't be put in a variational formulation. Oleg Alexandrov (talk) 04:22, 7 July 2008 (UTC)
All these things you mention can be put in a variational form. All you need to put something in a variational form is the ability to take the inner product over the necessary domain. If this is not possible FDM will not work either. -Art187 (talk) 14:07, 7 July 2008 (UTC)
How about ${\displaystyle u'=u.\,}$ Oleg Alexandrov (talk) 18:51, 7 July 2008 (UTC)
${\displaystyle a(u,v)=(u'-u,v)}$, umm this example shows me that you don't use FEM. -Art187 (talk) 20:46, 7 July 2008 (UTC)
Or ${\displaystyle u_{t}=u_{xx}\,}$ (heat equation), with finite element in both space and time. I don't think you can do finite element in time, at least I've never seen it done that way. Oleg Alexandrov (talk) 18:52, 7 July 2008 (UTC)
The heat equation can be done with finite elements in both space and time, according to Johnson's book (MR0911477), by using discontinuous Galerkin in time (see also doi:10.1137/0728003). I don't think that's a popular approach, but it is possible. -- Jitse Niesen (talk) 19:24, 7 July 2008 (UTC)
It's not popular because if you go to 3D space, then you have to be able to mesh 4D. -Art187 (talk) 20:46, 7 July 2008 (UTC)
I indeed did not use finite element except for rather simple problems.
So far I've been convinced that a large class of problems can be solved using finite element, even if it not always makes sense practically (e.g., for the heat equation using FEM for both space and time increases the dimensionality of the problem without any benefits).
However, I still regard as dubious the claim that any problem that can be solved with finite differences can also be solved with finite element. You need the bilinear form ${\displaystyle a(u,v)}$ to be positive definite and continuous, and I don't think you can always find the right spaces for that to happen.
What would you do for ${\displaystyle u''=\cos(u+u'),\ u(0)=a,\ u'(0)=b}$? Oleg Alexandrov (talk) 04:15, 8 July 2008 (UTC)
It is the same as anything else you take the inner product. The Dirichlet boundary condition is handled by eliminating degrees of freedom and the Neumann either through integration by parts or by adding another term in the form assembled only on the boundary. Whether you find it dubious or not it is mathematics, arguments not beliefs are important. You have already said you have only used the simpliest methods for FEM. Also the system does not need to be positive definite or continuous. -Art187 (talk) 07:16, 8 July 2008 (UTC)

At the end of the day, it is still not correct to say that finite difference method is a type of finite element. The finite difference method has its own derivations, its own analysis, and its own proofs of convergence. At most, you can say that a large classes of problems that can be solved with finite difference can also be solved in a finite element framework. And even this statement would need some good references. Art187 did provide some examples above, but a statement of such generality would need good references to back it up. Oleg Alexandrov (talk) 16:10, 8 July 2008 (UTC)

The book Numerical Methods for Partial Differential Equations By G. Evans, J. Blackledge, P. Yardley has its 4 chapters on FDM before introducing FEMs in chapter 5. So it might a good source to use here. I haven't actually read this article, and I'm probably not going to do that anytime soon... because even shorter ones on these topics are pretty badly written around here. Some1Redirects4You (talk) 16:07, 27 April 2015 (UTC)

## Issues with the rewrite

In the current version of the article, the section "Outline of general mathematical framework" suggests that finite element problems are always solved in ${\displaystyle H_{0}^{1}}$ which is not right. Oleg Alexandrov (talk) 04:38, 8 July 2008 (UTC)

Yes there are many problems still lurking in this rewrite. I kept the ${\displaystyle H_{0}^{1}}$ notation because it was suggested by other authors in the past. Here are a few other problems:

• Need to define what constitutes as a discretization of the domain.
• Probably should remove integration by parts or explain it.
• Need to give more examples of finite elements (I would suggest Raviart-Thomas since it is widely used and fairly easy to explain)
• Break out the assembly algorithm more.
• Discuss application of boundary conditions.
• Discuss mixed methods (methods that use multiple approximation spaces)
• Finish out the rest of the various finite element method types
• Put FDM and FVM back in.

As you can see this article can get quite long and should probably serve as a jumping off point to FEM not something that is inclusive. There has been talk of making a Mathematical treatment of the finite element method to go along with Engineering treatment of the finite element method. I would suggest doing so and here we can show simple concepts such as coercivity, h-,p-,k-convergence and so on. -Art187 (talk) 07:28, 8 July 2008 (UTC)

Finite element is a technical subject, that is unavoidable. That being said, the previous version of this article was a gentler explanation of what was going on than the current one. The current version skips over a lot of details, particularly omitting the 1D derivation of the variational formulation via integration by parts, which, I believe, is the key to getting people to understand how you go from a PDE to a variational formulation.

Unless there are objections in the next few days, I plan to replace the current section "Outline of general mathematical framework" in the article with what was there before, while keeping the rest. That won't be such a big change since what is there now is obtained by cutting things out (things which were important). Oleg Alexandrov (talk) 06:57, 10 July 2008 (UTC)

This page should not give all technical details of FEM, that place is in a book on the subject not an encyclopedia article. Things were cut out to only give a flavor of the method not to give a rigorous definition. With the number of complaints on this talk page about the technical issues, I would have to say I completely disagree with just using the old version of this page. -Art187 (talk) 09:20, 10 July 2008 (UTC)
The problem isn't that it's too technical, the problem is that it's awful. Really, there are all kinds of things on Wikipedia that are much more technical than FEM but more important; however this article has been totally useless to me personally and I had to go the to library to read books to learn FEM, going to the library for books is something I haven't done since 1995 or so, and I hope I never do it again. I think that this article needs a near complete rewrite, and I think that part of the problem is the blur between FEM (a method for solving weak equivalents of PDEs) and FEA (engineering practice). I really believe that in practice they're very different. — Ben pcc (talk) 23:26, 19 July 2008 (UTC)
There are things I like in both versions, but also things I don't like. I think that a concrete one-dimensional example is very helpful in giving the flavour of the method, so I'm sorry to see it gone. On the other hand, I think the new version flows better. I see that you two disagree on what version is more technical; I guess they're technical in different aspects.
I'd prefer to have, after the current "History" and "Application" sections, first a concrete example (along the lines of the old version) and then a more general discussion (along the lines of the new version). The two-dimensional example in the old version can go as far as I am concerned, or perhaps moved to a different article (finite element method in two dimensions). I would slightly generalize the general discussion, replacing H^1_0 by an arbitrary Hilbert space U, etc., though of course we should not try to give a rigorous definition of FEM (I don't think it's possible to catch all finite element methods in one definition). -- Jitse Niesen (talk) 11:34, 10 July 2008 (UTC)
Well, Art187, you removed the 1D example, but you still kept is discretization in the section on Finite_element_method#Choosing_a_finite_element, so now it is very unclear what that has to do with anything. I still believe going back to the original text is a better initial guess.
Also, the technical complaints were before the version you overwrote was put in.
I do agree that all the details are not necessary, but cutting the simplest 1D explanation is not the way to go. Oleg Alexandrov (talk) 15:01, 10 July 2008 (UTC)
I also support putting back the 1d example. Loisel (talk) 20:27, 10 July 2008 (UTC)
Actually, I think the article is getting shittier. I don't know what happened, I haven't been paying very close attention, but the "Outline of general mathematical framework" starts off with nonsense bilinear forms, where the notation has not even been explained, and also talks about basis functions for spaces that have not been described. I don't like the article. Loisel (talk) 20:30, 10 July 2008 (UTC)
I see now that the article got very bloated over time. I think that the article was better in 2005. http://en.wikipedia.org/w/index.php?title=Finite_element_method&oldid=33310404 Loisel (talk) 20:37, 10 July 2008 (UTC)
I copied back the mathematical discussion which was here before the merge, as that contained essential material to understand what was going on, and the merge was not carefully done. We can discuss where to go from here. Oleg Alexandrov (talk) 05:23, 11 July 2008 (UTC)

## Application of FEA

this method is most important in engineering especially in CAE with software like ansys and abaqus. it should not be in physics. —Preceding unsigned comment added by Saeed.Veradi (talkcontribs) 04:44, 20 July 2008 (UTC)

I agree physics is not the most relevant category. I removed it. Oleg Alexandrov (talk) 02:37, 22 July 2008 (UTC)

## Removed Half-truth in the difference between FE and FDM

The first point on the FDM was as follows

• The finite difference method is an approximation to the differential equation; the finite element method is an approximation to its solution.

Both methods are used to approximate the solution of a differential equation: that's what they are for. Although the author of the sentence probably has a good and well-established idea about both methods, this senence does not convey it. To me it sounds like: use FE if you want the solution of a differential equation. Use FDM if you want the differential equation itself. I hope we can agree that that is definitely not true. —Preceding unsigned comment added by 130.89.67.43 (talk) 16:36, 22 April 2009 (UTC)

## Variational form

The technical section of the article mentions variational form with a link to the calculus of variations. The link is not appropriate because the linked page does not define variational form. Jfgrcar (talk) 22:29, 7 December 2010 (UTC)

The introduction links to "Variational method" which really is a seemingly unrelated article on Quantum Mechanics. — Preceding unsigned comment added by 2607:EA00:104:3C00:21C9:8DF1:4B1A:451 (talk) 22:11, 12 March 2013 (UTC)

## How is the gradient and the Laplacian calculated?

I have a question about this method, for which I think the answer should be in the article (which it currently not as far as I can see). I wonder how the gradient and the Laplacian of the scalar field is calculated, after the discretization has been carried out? This is highly relevant to the article, since being able to calculate these two is necessary in order to be able to solve most PDE:s at all using the finite element method. --Kri (talk) 20:42, 6 April 2011 (UTC)

## Should the title be hyphenated?

Why is it "finite element method" instead of "finite-element method"? The first means "finite method of elements", whereas in reality it is "method of finite elements" and thus in the adj. + noun form should be "finite-element" + "method". — Mikhail Ryazanov (talk) 02:31, 6 February 2014 (UTC)

By convention it is not hyphenated. There are many similar expressions without hyphens. The hyphen is only used to avoid misunderstanding, which is unlikely in this case. −Woodstone (talk) 09:38, 6 February 2014 (UTC)
By which convention? English_compound#Hyphenated_compound_modifiers says that numbers and other quantitative modifiers (such as "high", "low", "full"...) must be always hyphenated. I believe, "finite" falls in the same category. — Mikhail Ryazanov (talk) 02:44, 7 February 2014 (UTC)
I'd say the convention according to what is used in the literature. A GScholar search for the term "finite-element method" yields the first three pages of 29 hits with the method named without a hyphen and 1 hit for the method named with a hyphen. GBooks yields a similarly lopsided ratio. Per WP:COMMONNAME, we generally use the most common form of the term in the literature. Since both forms occur, however, I've added a redirect from Finite-element method --Mark viking (talk) 03:42, 7 February 2014 (UTC)
It seems that this field uses a special variant of English. :–) Otherwise, I don't know how to explain phrases like "open source finite element software programs" (yes, without any hyphens and with mysterious "software programs", which probably means "computer programs" or "software") and why "Runge-Kutta", "Euler-Bernoulli" and "Navier-Stokes" are written with hyphens instead of dashes. — Mikhail Ryazanov (talk) 22:16, 7 February 2014 (UTC)
Finite-element method is proper, but I don't think finite element method is improper. I'd prefer finite-element method, but the common usage is without the hypen.
Endashes are the WP house style. Runge-Kutta and Navier-Stokes (hypen links) redirect to articles with endash between the names. Glrx (talk) 21:26, 8 February 2014 (UTC)
I agree that this a matter of debate, it basically depends on the most common usage. Another example, "short circuit" and "short-circuit". 137.132.22.191 (talk) 03:20, 19 February 2014 (UTC)

I agree with Mark Viking on this. I haven't seen this term hyphenated much in the literature. The people who spend their time enforcing some imaginary laws of prescriptive English... should probably find something else to do. Some1Redirects4You (talk) 16:04, 27 April 2015 (UTC)