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# Why not use FDM in solid mechanics?

Finite Difference Method (FDM) and the related techniques such as FVM, are often found put to great use in fluid mechanics. See any simulation showing not only streamlines but also vortex shedding, turbulent mixing, etc.

Yet, when it comes to solid mechanics, Finite Element Method (FEM) is most often the method of choice. Actually, FEM is probably the *only* computational method used in solid mechanics. Most books on solid mechanics and structural analysis do not even mention FDM. A few that do, restrict FDM only to the Laplace's equation and the bi-harmonic equations--not to the general stress analysis problem in 3D.

Why is this so?

If you really think hard about it, you can see that the usual arguments forwarded in favor of FEM and against FDM really do not hold--not at least to the extent this is routinely supposed.

For example, consider the one big advantage usually ascribed to FEM, namely, its ability to handle complex geometries and non-rectangular meshes.

But if the first reason (complex geometries) really is the cause, then why does FEM formulation of contact stresses require a special treatment?

And what is the true relevance of the second reason (non-rect. meshes of FEM) in this day and age of abundance of memory? You can always use a little finer mesh, esp. near curved surfaces, isn't it? In any case, shouldn't it be just a matter of a little more theoretical development of FDM? (Think FVM and the related developments here.)

So, it should be possible to use FDM in Solid Mechanics--especially because FDM is so simple. Yet, it is not. Why not? Is it just a matter of predominant culture in research since the 1960's?

You are welcome to post all your thoughts. However, please let them be really well thought out. Above all, please do not write under the impression that I do not like or use FEM. I do. Also, do not write to advice me where to really pick up FEM from. I think I know that too...

It's just that I can see acertain clear conceptual advantage with FDM that is absolutely not present with FEM.

Thanks in advance for your *well thought out* replies. In particular, thanks in advance for *not* just reproducing the same opinions you heard or read in your graduate courses on FEM, numerical analysis, or solid mechanics. Thanks for *that*, really!

## FEM vs FDM

R. Chennamsetti, Scientist, India

Hi,

I went through your post. I gave a though on that. The following are my thoughts.

[1] In FDM we estimate the dependent variables at grid points. we don't use any interpolation concept. But, in FEM we use interpolation function, we evaluate at nodal points and the value of dependent variable can be obtained, by interpolation, any where in the element. But, in FDM, we have to do this interpolation manually. In FEM this is inbuilt in the formulation.

[2] In FDM, at irregular boundaries, we need to use more number of cells (grid points). But, in FEM simply use a higher order element with the same number of elements. This is more easier. In FEM, irregular boundaries can be modeled easily than in FDM.

[3] When dynamic analysis is done, in time domain the discretisation is still done in FDM only. FEM compuations are more expensive in this case. (There is a good book on FD for dynamics).

[4] One reason why FEM is more popular is the commercial softwares. Most of them are written in FEM.

FDM can also be used for solving Solid Mechanics problems.

- Ramdas

## Re: FEM vs. FDM

Hi Ramdas,

About your point # [1]: It is true that in FDM the evaluation occurs only at the node points. But the discrete model is also *exactly* solved for at these finite number of points. In contrast, in FEM, we settle for a solution that is evaluated everywhere (at infinitely large no. of points) within an element, but it also is a solution that is in general *wrong (inexact)* at each and every one of all those points--only an overall measure for the element is solved for exactly. ... To conclude, in any case, FEM is not as clear a winner as it is routinely made out to be. Then, why not use FDM? That's the simple question I ask!

About your point # [2]: I am not sure about the point you make here. If it were that easy to match a curved boundary with a function passing through a few (nodal) points, life for computer graphics programmers (and NURBS researchers) would have been a lot easier. It is not. Implication: Issues related to boundaries, contact, etc. remain (the way I touched already). Another thing. Higher order elements anyways do add computational complexity.

Thanks for pointing out your point #s [3] and [4]!

On point # [3], I would like to know about the books on dynamics using FDM you have in mind.

I myself wanted to make point # [4] too. IMHO, it just shows that there would be a kind of circle here, for such a prevalent use of FEM in SM (as against in FM): There is research money --> So, there's technology --> So, there's software --> So people use it for research --> So, people ask for research money --> Back to square one. "Justifications" often times reduce to something as simple as that--not necessarily to technical superiority (as is often made out, in academic circles / text books).

Thanks, anyways, for sharing your thoughts!

## Irregular meshes

Another reason to use FEM is that it is much easier to handle irregular meshes. The finite element method -- mapping to the parent domain etc makes it so much easier to have a general method, as compared to the FDM

BTW, since you commented on NURBS, you may be interested in this:

www.ices.utexas.edu/research/reports/2006/0604.pdf

-Amit Ranade

## Re: FEM vs FDM or why use different numerical methods

Over the years I have had the opportunity to work with a number of numerical methods. The choice of numerical method depends on the problem that is being considered.

The finite difference method (and its finite volume counterparts) are used widely in computational fluid dynamics (CFD). The main reason is that a fluid is better modeled in an Eulerian frame. Even so, applying certain boundary conditions can be a nightmare - for example when an eddy leaves a domain at a boundary getting the vortices correct is nontrivial. Also there is some numerical diffusion when such methods are used which can be neglected in most CFD computations.

Solids are better modeled in a Lagrangian framework and the finite element method fits better into such a framework under some circumstances.

Finite difference methods have been used for solid mechanics in the past - particularly for hydrodynamic simulations (see M. L. Wilkins, Computer Simulation of Dynamic Phenomena) and also for poroelastic simulations for rock mechanics (I can't recall the name of the software at this moment but it's used by Call and Nicholas for all their simulations). The main problem that is faced when one tries to used FDM methods for solids is that flux boundary conditions can be difficult to implement (and zero flux boundary conditions do not come for free). Also, the code can get hairy when you try to implement irregular grids and also have options for various orders of approximation of the differential operators.

I don't know the history of the choice of FEM very well - perhaps some of the experts here can elaborate. But I feel that ease of implementation was what first made the method popular. That does not imply that FEM is the only game in town. It is just convenient for a particular set of problems that is of interest to mechanicians. I am yet to find one method that can solve all problems of mechanics better than all other methods.

## A good reply from Biswajit, + many other points...

Dear Biswajit,

A nicely informative and well-balanced reply, containing many new relevant points/observations! Thanks!!

Here are some comments (on your reply), and also some additions (in general)...

First, the comments on your reply:

The general ease of FEM in handling boundary conditions, as compared to FDM, seems a valid point. Though this point goes in favor of FEM, it would strike one only if one actually has considered FDM as an option! I had thought of it.

But the idea of Eulerian and Lagrangian had completely escaped me! Thanks for introducing this theoretically very illuminating idea in this debate.

Another thing. Do you mean to say that though not in FDM, it is easier in FEM to handle vortices crossing boundaries? (I am not knowledgeable here.)

The issue of numerical diffusion *seems* debatable on the face of it. If it arises in one form of matrix-based model, why doesn't it arise in another? Especially, if the matter of concern was the closed domain BV problems, not the initial value propagation problems. Note, practically speaking, for engineering solid mechanics, the former problem is what matters. So, I would guess that at least for practical SM, numerical diffusion would not be a show-stopper. Of course, I don't know, haven't thought too deeply about it, and could be wrong... Explanatory comments are welcome.

The other new points about FDM that you nicely bring up: flux boundary conditions, irregular grids, the lack of any guiding rule to choose which optional form of DE one should pick up for discretization or modeling.

All these points are very relevant, but IMHO, they perhaps can be taken as peculiarities of the technique rather than its limitations.

Here, I want to point out how the research community treats BEM in SM. BEM, too, has its own peculiarities, but it still gets researched, commercialized, and used in practical SM problems--all of which is unlike FDM. (I don't want to go back to the lionization of mathematics issue, but doesn't it *really* seem like BEM gets funded and highlighted precisely *because* it's more complicated than FDM is, from a mathematical viewpoint?)

About irregular grids. Do you mean non-uniformity (an issue of mesh size, i.e. finer mesh at some places) or do you mean the desirability of a non-rectangular and non-polar sort of grid (an issue of shape)? If the shape is the point, I think it is a non-issue. It doesn't matter how you split the domain so long as you get good results. As far as non-uniformity of size goes, it seems to me that even if you were to selectively refine a rectangular grid and directly enforce matching of variables at the boundary between the dense and coarse regions by direct linear interpolation of the unknown variable, it should work out acceptably well. After all, FDM is nothing but finite differencing--linear interpolations--across points. And, from a software development point of view, for enforcing the match, writing such modules would be relatively easily possible. The computational geometric problem here is much easier to address than the one involved in adaptive refining of mesh, and the latter is these days a routine feature of many meshing software, whether the s/w starts from a CSG-based or a BREP-based geometric model.

-----

Now, my additions. As usual, fairly long. :)

History-wise, I gather this much about FEM: There was Courant (1943) sitting in his mathematical isolation. So, nothing happened apart from the publication of an article or so. Then, there was Argyris, but he was across the Atlantic (and so, he was almost non-existent), and he was in academia (and so, he was completely non-existent). So, the only way left to explain why FEM came out of its shell is to say that Boeing played the mother-hen. Initially, engineers modeled a continuum solid as a truss! They soon enough went past that stage but stayed mostly linear. The days of cold-war and of increasing state-funding for American science were just beginning right at that point of time. That partly explains why FEM came to become so very entrenched in USA--and hence in the world. (In any government-bureaucracy, as against in free markets, anything related to ideas has a tendency to stay put as-is, and anything related to government-expenditure has a tendency to grow.) It took time for the new technique to reach to the FM community. The codes in FM were already being written using FDM. To justify expenditure on those codes, FM kept defending FDM. This gave it a chance to grow. ... I am not knowledgeable, but have been mostly guessing here.

The ease of implementation of FEM can't be the reason for its growth as anybody who has implemented both FEM and FDM would know. In fact, even FEM people grant that much to FDM (right in their own text-books too!). So, looks like Boeing's patronage was the main reason the new technology got the necessary *initial* push. (Listen, Pune industry, will you?) Both mathematicians and bureaucrats came much later. (LOL!) I think it's good for FEM that the "innovative" and "creative" Sand Hill vC's have yet to arrive the scene! I hope they don't. (See, I don't at all wish ill of FEM!)

-----

OK, coming back to the things technical, here are some additional notings:

(i) Sir Richard Southwell invented the Relaxation Method (RM) in 1930s. RM is a sort of like manually done FDM. Southwell's book (pub. 1940s) shows how his group had applied RM to the design of, among other things, a connecting rod, during the war-time effort in England (the II WW). This is a very direct application of an FDM-like technique to a practical problem from *solid mechanics*.

(ii) When Southwell visited USA in (I suppose) 1950s, Timoshenko was so impressed with the new technique that he immediately included an appendix on FDM in the next edition of his book on elasticity. (Also see Zienkiwicz' address available on iMechanica.) In the appendix, Timoshenko solves only a beam-theoretical and a Laplace equation problem, but not a more general plane stress problem, let alone a completely general 3D stress/strain problem.

In short, Southwell's book of 1940s contains a little more advanced problems of computational elasticity than Timoshenko's book of 1950s does--even if today almost no one knows about RM.

(iii) Ghali, Neville and Brown's book on Structural Analysis, in its first edition (1972), used to have three chapters on FDM, I gather. There used to be an illustrative plane stress problem done up using FDM. In the subsequent editions, they cut down on FDM. In the recentmost 5th ed., FDM has been cut down to just one chapter, and FEM has been expanded to two chapters. In my private correspondence with him, Prof. Ghali indicated that this was done to cover modern techniques and to keep the book-size down as per publishers' requests.

Very similar to Ghali et al's current edition, Boresi and Chong only state, but do not illustrate in computational detail, how to solve a plane stress problem. Neither book deals with a 3D problem.

A few other books do not even abstractly state how to solve the plane-stress problem; they only illustrate the Laplace or the bi-harmonic equation. Such books include, e.g., Solecki and Conant.

Most other authors--of SM or elasticity--remain blissfully unaware of the possibility of using FDM in SM--or at least keep their readers so.

Hence, this post at iMechanica!

(iv) As Biswajit says, a few research papers use mixed photo-elasticity and FDM. One stream of development uses photo-elasticity to find out stresses at the boundaries and feed that in as constraints in the numerical model--thereby circumventing the BV specification problem, in a way! For example, papers by K. Chandrasekhara and K. Abraham Jacob. (I am still trying to trace/obtain their papers.)

(v) One new approach is the "Cell method," developed by Prof. Tonti and his PhD students (e.g., Francesca Cosmi) in Italy. This approach is more like FVM for elasticity. They have even applied it for a simple problem of elasticity of poros materials.

-----

Special thanks, again, to Biswajit, for a very nice reply. Practically, his point that no one tool/technology will be equally satisfactory for all applications is right on target. But what I most appreciated was the great theoretical point he introduced in the discussion: Eulerian vs. Lagrangian. (I worry a lot about fields and particles.) I will come back on that one sometime later on.

In the meanwhile, I would welcome it if anybody could please cite research papers / technical reports / book chapters not mentioned above--i.e., literature resources that apply FDM to a *general* plane stress or plane strain problem, or, better still, to a general 3D stress analysis problem (but not to the routinely given Laplace/bi-harmonic/simple beam deflection problems). Thanks in advance!

## FDM is a special case of collocation methods

Finite difference method (FDM) is only a special case of the generic collocation method. If we use the weights as Dirac's delta functions and write the weak for as a point collocation, we get identical form as that one gets in FDM. Please refer Cook et.al. for further details.

## A "special case" of the physically arbitrary generalization(s)!!

1. I knew the result you mentioned. (Dirac delta weights in point collocation equivalent to FDM). I still wrote my post. But then, you still went ahead and wrote your reply the way you did.

So, now I have no option but to ask *you* in particular: Would it be possible for you to give a physical explanation of the "general" result you speak of?

Note, you must give the physical explanation *in the general case*--i.e., your *physical* explanation must remain valid even outside of the very very special case from solid mechanics: the reversible linear elastic solids (and I can't be sure if I didn't miss an applicable qualification or two here), where both FEM and FDM happen to be equivalent.

Can you give a physical explanation of the general WR/point collocation method? In your own words? If you attempt to do that, it will become possible for me to point out precisely where and how the general principle you mention differs from FDM--how it no longer *can* at all include FDM. (My confidence rests on the mathematical difference between the strong and weak forms. The difference will stay, in the general case, through any form of discretization.)

BTW, no one has been able to do that--to give a physical explanation of the generalized theory of FEM, whether you describe it using a functional or a residual form, or any other. That's the reason I call such "generalizations" physically arbitrary. Also see my post on Lionization of Mathematics: http://www.imechanica.org/node/1440

2. Now, if you cannot give a physical explanation *in the general case*, then what compelled you to make it look as if the FEM research/results *include* FDM research/results? They do not. Please correct me if I am wrong here. But while correcting, do provide your own arguments--do not simply quote authority.

## FDM vs. FEM

Dear Ajit Jadhav,

Thank you very much for this interesting topic. I thought it is a good idea to ask this question to one of the first FEM programmer in the world. His name is Hussein Kamel and he is a professor emeritus at the University of Arizona. Here is the response that I got from him:

"The discussion of FDM vs. FEM in solid mechanics took place a long time ago, and was

settled to most peoples' satisfaction at that time in favor of FEM. The reasons may be

summarized simply as follows:

1. FEM can handle complex geometry much more effectively.

2. Attempts to modify FDM to address complex geometry resulted in something very similar

to FEM, without the elegance of FEM

3. Some researchers (e.g. A.K.Noor) proved that the two methods can be mathematically

related.

4. FEM is a natural companion to solid modeling and other increasingly popular

computer-related disciplines.

5. Finally, nothing succeeds like success.

FDM is still useful in the time domain (e.g. time integration), and in some disciplines

other than solid mechanics."

Regards,

Erkan Oterkus.

## Objections against FDM in SM, re-examined...

Hi Erkan,

Thanks for your interest and comments. ... The topic certainly is interesting, but I guess putting it as the "FDM vs. FEM" issue makes it a little bit spicy too :) ...

I respect prior discussion. ... Actually, I respect prior FEM researchers (and their researches involving applications of FEM and about FEM itself) far more than their arguments for the *superiority* of FEM over FDM.

The superiority arguments simply do not hold the way they are supposed to--that's one of the overall points I try to make here.

Coming to the specifics of your ponts.

Most of Prof. Kamel's arguments have already been discussed but let me reiterate the replies. There are finer shades of differences that may perhaps better come to light in re-writing the replies. First, I answer with same point numbers as you have used above.

1. CFD has an equally or even more accute need to handle *curved* boundaries, and still, FDM has been such a success in fluid mechanics. If so, why does this point still arise in solid mechanics?

2. FEM may look elegant on the face of it. But outside the applicability of the stationarity principle (whether this is stated in the functional or the WR form) its formulation remains, from the physics point of view, very much arbitrary. This means: Nobody--neither FEM researchers nor I nor anyone else I know of--knows what physics to give for the *general* WR/functional formulation.

If FEM weren't an "approximate" (i.e. numerical analysis) technique, its physical arbitrariness would have been "found out" far earlier.

Come to think of it, what use is mere elegance if the entire construction floats in thin air? None addresses (or has ever addressed) this real burning issue. Why not? (See, again there is that lionization of deductions, abstractions and mathematics issue I discussed earlier in my track!)

Another point. Not every attempt to generalize FDM-like approach *can* possibly result in something like FEM, because the two conceptually differ so much. See the point 3. below. Also, note the remark about Prof. Tonti's "cell method" noted above. This method is more like FVM than like FEM. In older times, people didn't think of such possibilities.

3. Conceptually, FDM is the approach based on differential calculus and FEM is the approach based on the integral calculus. You cannot erase this difference. So, you can only think of establishing relation. Within this fundamental classification scheme, there are two levels at which the question of the validity of a supposed relation can be examined.

(a) The two can be said to be very directly related to each other via the Fundamental Theorem of Calculus, but it is valid only for the continuum case, i.e. when the mesh is infinitely fine.

A relevant aside: Typically, students are taught this point (3(a)) in the following, disconnected manner, reinforcing arbitrary biases... In undergraduate Numerical Analysis courses: "FDM has been proved to converge to analytical solution." Separately, in the graduate FEM courses (without any mention of FDM): ".... Thus, FEM can be proved to converge to analytical solution." Still separately, in *any* course: ".... So, FEM is superior to FDM!!"

(b) For *finite* sized elements/mesh, the two can be *directly* related only in that one very special case: the case of the reversibly linear elastic etc. solids (in the small deformation approximation).

Now, if a very indirect relation is sought, the two techniques could always be indirectly related, via, say, some or the other series expanision, thousands or all of whose terms are significant. Now, *that* kind of a "relation" is as bad as being non-existent...

So, for finite sized meshes, in the general case, the two cannot be said to be directly related, i.e. there is no first or second order relation in the general case. That's precisely why I say that the general WR technique is physically arbitrary. I wouldn't have said so if a direct relation existed.

4. This point is worse than being plain wrong. FDM already is (and has been) an equally natural companion. Just have a look at the CFD codes and results. (BTW, why does solid mechanics community behave so ostrich-like, refusing always to learn anything from CFD)?

5. So, the issue is not so much FDM vs. FEM. It's more like: Success of FDM in Fluids vs. Success of FEM in Solids. The question is: Why? (LOL! The answer is: because one is mathematically more challenging and so, for those who are so inclined, FEM provides an easier platform to take the flights of fancy made from floating abstractions. That's why!)

-----

To add:

6. At the risk of boring you with repeatation, but only in the interest of clarity, let me say: The issue is *not* FEM vs FDM. It is: Why not use FDM in SM--esp. if FDM is such a success in CFD.

7. What is happening here is that people are spared being forced to come to terms with the physical arbitrariness of the General WR/Functional formulation because "everyone knows" the models have to be compared to physical experiments anyways. With that final proviso always there, acting as the real savior, the actually serious malady which exists with the theorization of the FEM technique, has always found a kind of camouflage. Perhaps nice and comforting. But not in the best interest of the FEM theory itself! (or of its applications!!)

8. Finally, one last point. This is not an appeasement of the FEM community but an honest statement.

(a) I do think FEM is valuable. But not to the extent that SM FEM community thinks and makes out it is. And, the approach is too indirect to be really called elegant. The continuum-case variational principle, by some imagination, can be thought to be looking elegant to some eyes. But certainly not the general WR/Functional-based formulations. And it is the general formulation which is truly relevant for general PDE solutions.

(Personally, I won't call even the *simple* variational principle, as expounded in the linear case, to be *very* elegant because it's not so direct a way to look at physical reality. But yes, at least some people, right from Fermat onwards and including Nobel laureates like Feynman, have found thinking in terms of variational formulation very elegant. But the problem with this "elegance" is its easy susceptibility to unphysicality. Thinking on the variational lines, some of the best people, e.g. Feynman, also have posed models of *physics* in which "particles" from *future* travel back in time in such a way as to precisely "arrive" "at" a point, and, lo and behold, vanish right then and there! (LOL!!) The absurdities result only because they would rather give up common sense, even physics, but not an "elegant" mathematical principle like a variational statement.

(b) I am far more impressed by the physical->mathematical mapping done by the active FEM researchers. Consider, for example, transients (in thermal engineering or otherwise). Or, composites. Or, fracture. Virtually, any kind of complex modeling problem.

The way I look at it, precisely because the technique (FEM) is so complicated that I say it must require far more ingenuity to create workable mathematical models that can use FEM in simulating such advanced features as those mentioned just above (transients, dynamics, etc.)

Yet, I don't necessarily always find such ingenuity very admirable. (When in an especially bad mood, I could even call it a special variety of stupidity--because the technique the ingenuity is so bent upon using is so impractical for the purpose at hand--that's why!!)

Yet, it also is a fact that so much recent progress in computational solid mechanics has FEM applied in it. The mapping or the progress of *conceptual* modeling, from the raw physical observations to mathematical modeling to a *pre-FEM* modeling stage, found in such solid mechanics literature, is oftentime very much admirable--even if it has been put out by the same FEM community (and even if some of them are outright biased against FDM).

This part of their work *is* non-arbitrary, and often very impressive. (That's probably why people are amazed when one says that the idea of FEM in SM is so twisted--people typically mix up the two issues. They project the admiration they have for the physical-mathematical modeling on to the technique of FEM itself.)

I begin to have problem once, having so nicely isolated the physical factors of a modeling situation, people mangle it so willingly to put it in the specifically WR/functional form terms.

So, the generalized WR technique (outside the reversible linear elastic etc. case) itself is nowhere as admirable.

-----

On a personal note: Erkan, feel free to call me Ajit.

## Concluding "FDM in SM" for the time being...

I got excited about Lagrangian and Eulerian viewpoints more for what the issue seemed to suggest towards theoretical development of my research in general terms. But coming to the specifics of FDM in SM, let me note the following points here.

For the *continuum* case, in the small deformation theory, the descriptions from the Lagrangian and the Eulerian viewpoints are identical. So the issue (L vs E) only arises for large deformations such as those encountered in plasticity, metal forming, creep, dynamic loading, fragmentation, etc.

For *FDM* too, for quasi-static problems and small deformations, the issue of Lagrangian vs. Eulerian doesn't really matter much. From practical design engineering point of view, this category of problems constitutes the 80% (or the baseline) case--it basically covers almost all of Shigley, for example.

FEM does provide for a uniform formal structure to accomodate specifying a variety of boundary conditions. Such a structure has not been developed for FDM. So, ease of BC specification is certainly an advantage with FEM. But even here, note, in building FEM models, we often specify BCs with a practised ease but our procedures still involve ad-hoc assumptions coming from outside of the formal FEM theory. For example, in SM, we allocate to nodes what actually is a continuous surface traction. With due development it should be possible to evolve similar norms in the handling of the BCs while using FDM too.

So, to conclude this debate for the time being, unless someone adds a really good *and* new point later on, let me state this much: I can't think of any objection that selectively applies only to FDM in solid mechanics, but not to other combinations of a method and an area of application: FDM in fluid mechanics, FEM in solid mechanics, BEM in fracture mechanics, etc.

## Why FEM is not used in CFD

I just came across this nice discussion, even though it was posted a long time ago.

In my opinion you are quite critical with respect to the "complex geometry" advantage of FEM. FEM is very good at handling complex geometries because it can handle triangles. FDM is very awkward with triangles. FDM can handle complex geometries by using generalized curvilinear/boundary fitted coordinates, by variable grid sizes and by local grid refinement, but none of these are easy to do.

Then why is FEM not used in CFD? Simply because directional derivatives are difficult on an unstructured grid. This concerns transport phenomena: advection processes. Advection is discretised much easier on a structured (i.e. locally orthogonal) grid, particularly when you want high accuracy and be able to capture shocks as well (e.g. using flux limiters). Probably FEM-people have found appropriate ways to deal with advection too, but not as powerful and simple as it is done in FDM.

FVM methods are somewhere in between and combine some of the advantages and disadvantages of both. You can use rectangular volumes in large parts of the domain and polygonal volumes near boundaries or internal interfaces. At the expense of complex administration and complex approximation schemes, you may get flexibility *and* accuracy.