Finite element method: Difference between revisions

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{{Differential equations}}

The '''finiteFinite element method''' ('''FEM''') is a popular method for numerically solving [[differential equation]]s arising in engineering and [[mathematical models|mathematical modeling]]. Typical problem areas of interest include the traditional fields of [[structural analysis]], [[heat transfer]], [[fluid flow]], mass transport, and [[electromagnetic potential]]. Computers are usually used to perform the calculations required. With high-speed [[Supercomputer|supercomputerssupercomputer]]s, better solutions can be achieved, and are often required to solve the largest and most complex problems.

The FEM is a general [[numerical analysis|numerical method]] for solving [[partial differential equations]] in two- or three -space variables (i.e., some [[boundary value problem]]s). There are also studies about using FEM to solve high-dimensional problems.<ref>{{Cite journal

| lastlast1 = Hoang

| firstfirst1 = Viet Ha

| last2 = Schwab

| first2 = Christoph

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| year = 2005

| publisher = SIAM

| doi = 10.1137/040606041030601077

| hdl = 20.500.11850/147656

}}</ref> To solve a problem, the FEM subdivides a large system into smaller, simpler parts called '''finite elements'''. This is achieved by a particular space [[discretization]] in the space dimensions, which is implemented by the construction of a [[Types of mesh|mesh]] of the object: the numerical ___domain for the solution, which has a finite number of points.

| hdl-access = free

The finite element method formulation of a boundary value problem finally results in a system of [[algebraic equation]]s. The method approximates the unknown function over the ___domain.<ref>{{cite book

}}</ref> To solve a problem, the FEM subdivides a large system into smaller, simpler parts called '''finite elements'''. This is achieved by a particular space [[discretization]] in the space dimensions, which is implemented by the construction of a [[Types of mesh|mesh]] of the object: the numerical ___domain for the solution, whichthat has a finite number of points. FEM formulation of a boundary value problem finally results in a system of [[algebraic equation]]s. The method approximates the unknown function over the ___domain.<ref>{{cite book

| title = A first course in the finite element method

| author = Daryl L. Logan

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| year = 2011

| isbn = 9780495668275

}}</ref> The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the [[calculus of variations]].

Studying or [[Analysis|analyzing]] a phenomenon with FEM is often referred to as '''finite element analysis''' ('''FEA''').

==Basic concepts==

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The subdivision of a whole ___domain into simpler parts has several advantages:<ref name=":0">{{cite book |first1=J. N. |last1= Reddy |title= An Introduction to the Finite Element Method |edition=Third |publisher=McGraw-Hill |year=2006 |isbn=9780071267618}}</ref>

* Accurate representation of complex geometry;

* Inclusion of dissimilar material properties;

* Easy representation of the total solution; and

* Capture of local effects.

TypicalA worktypical outapproach ofusing the method involves the following steps:

# dividingDividing the ___domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations tofor the original problem.

# systematicallySystematically recombining all sets of element equations into a global system of equations for the final calculation.

The global system of equations hasuses known solution techniques and can be calculated from the [[initial value]]s of the original problem to obtain a numerical answer.

In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often [[partial differential equation]]s (PDEPDEs). To explain the approximation inof this process, the finite element methodFEM is commonly introduced as a special case of the [[Galerkin method]]. The process, in mathematical language, is to construct an integral of the [[inner product]] of the residual and the [[weight function]]s; andthen, set the integral to zero. In simple terms, it is a procedure that minimizes the approximation error by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are [[polynomial]] approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally withusing the following:

* a set of [[algebraic equations]] for [[steady state|steady-state]] problems,; and

* a set of [[ordinary differential equation]]s for [[transient state|transient]] problems.

These equation sets are element equations. They are [[linear]] if the underlying PDE is linear and vice versa. Algebraic equation sets that arise in the steady-state problems are solved using [[numerical linear algebra]]ic methods. In contrast, [[ordinary differential equation]] sets that occur in the transient problems are solved by numerical integrationintegrations using standard techniques such as [[Euler's method]] or the [[Runge–Kutta methods|Runge-Kutta]][[Euler–Bernoulli methodbeam theory|–]]Kutta method.

In stepthe (2)second step above, a global system of equations is generated from the element equations by transforming coordinates from the subdomains' local nodes to the ___domain's global nodes. This spatial transformation includes appropriate [[Transformation matrix|orientation adjustment]]s as applied in relation to the reference [[coordinate system]]. The process is often carried out byusing FEM software usingwith [[coordinates|coordinate]] data generated from the subdomains.

The practical application of FEM is known as ''finite element analysis'' (FEA). FEA, as applied in [[engineering]], is a computational tool for performing [[engineering analysis]]. It includes the use of [[mesh generation]] techniques for dividing a [[complex system|complex problem]] into smallsmaller elements, as well as the use of software coded with a FEM algorithm. InWhen applying FEA, the complex problem is usually a physical system with the underlying [[physics]], such as the [[Euler–Bernoulli beam theory|Euler–Bernoulli beam equation]], the [[heat equation]], or the [[Navier-Stokes equations|Navier]][[Euler–Bernoulli beam theory|–]]Stokes equations, expressed in either PDEPDEs or [[integral equation]]s, while the divided, smallsmaller elements of the complex problem represent different areas in the physical system.

FEA may be used for analyzing problems over complicated domains (likee.g., cars and oil pipelines) when the ___domain changes (ase.g., during a solid-state reaction with a moving boundary), when the desired precision varies over the entire ___domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource, as they remove multiple instances of creating and testing complex prototypes for various high-fidelity situations.<ref>{{citationCite journal needed|date=March2022 |title=Editorial Board |url=https://doi.org/10.1016/s0168-874x(22)00118-4 |journal=Finite Elements in Analysis and Design |volume=211 |article-number=103845 |doi=10.1016/s0168-874x(22)00118-4 |issn=0168-874X|url-access=subscription 2021}}</ref> For example, in a frontal crash simulation, it is possible to increase prediction accuracy in "important" areas, like the front of the car, and reduce it in itsthe rear (of the car, thus reducing the cost of the simulation). Another example would be in [[numerical weather prediction]], where it is more important to have accurate predictions over developing highly nonlinear phenomena, (such as [[tropical cyclone]]s in the atmosphere, or [[Eddy (fluid dynamics)|eddies]] in the ocean), rather than relatively calm areas.

A clear, detailed, and practical presentation of this approach can be found in the textbook ''The Finite Element Method for Engineers''.<ref>{{Cite book |last=Huebner |first=Kenneth H. |title=The Finite Element Method for Engineers |publisher=Wiley |year=2001 |isbn=978-0-471-37078-9}}</ref>

==History==

While it is difficult to quote the date of the invention of the finite element methodFEM, the method originated from the need to solve complex [[Elasticity (physics)|elasticity]] and [[structural analysis]] problems in [[civil engineering|civil]] and [[aeronautical engineering]].<ref>{{Cite journal |last1=Liu |first1=Wing Kam |last2=Li |first2=Shaofan |last3=Park |first3=Harold S. |date=2022 |title=Eighty Years of the Finite Element Method: Birth, Evolution, and Future |journal=Archives of Computational Methods in Engineering |language=en |volume=29 |issue=6 |pages=4431–4453 |doi=10.1007/s11831-022-09740-9 |s2cid=235794921 |issn=1134-3060|doi-access=free |arxiv=2107.04960 }}</ref> Its development can be traced back to work by [[Alexander Hrennikoff]]<ref>{{Cite journal |last=Hrennikoff |first=Alexander |title=Solution of problems of elasticity by the framework method |journal=Journal of Applied Mechanics |volume=8 |issue=4 |pages=169–175 |year=1941 |doi=10.1115/1.4009129 |bibcode=1941JAM.....8A.169H }}</ref> and [[Richard Courant]]<ref>{{Cite journal |last=Courant |first=R. |title=Variational methods for the solution of problems of equilibrium and vibrations |journal=Bulletin of the American Mathematical Society |volume=49 |pages=1–23 |year=1943 |issue=1 |doi= 10.1090/s0002-9904-1943-07818-4 | doi-access=free }}</ref> in the early 1940s. Another pioneer was [[Ioannis Argyris]]. In the USSR, the introduction of the practical application of the methodFEM is usually connected with the name of [[Leonard Oganesyan]].<ref>{{cite web |url=http://emi.nw.ru/INDEX.html?0/resume/oganesan.htm |title=СПб ЭМИ РАН |website=emi.nw.ru |access-date=17 March 2018|archive-url=https://web.archive.org/web/20150930001741/http://emi.nw.ru/INDEX.html?0%2Fresume%2Foganesan.htm |archive-date=30 September 2015 |url-status=dead}}</ref> It was also independently rediscovered in China by [[Feng Kang]] in the laterlate 1950s and early 1960s, based on the computations of dam constructions, where it was called the ''"[[finite difference method]]" based on variation principle''principles. Although the approaches used by these pioneers are different, they share one essential characteristic: the [[Polygon mesh|mesh]] [[discretization]] of a continuous ___domain into a set of discrete sub-domains, usually called elements.

Hrennikoff's work discretizes the ___domain by using a [[Lattice (group)|lattice]] analogy, while Courant's approach divides the ___domain into finite triangular subregionssub-regions to solve [[Partial differential equation#Linear equations of second order|second -order]] [[elliptic partial differential equation]]s that arise from the problem of the [[torsion (mechanics)|torsion]] of a [[cylinder (geometry)|cylinder]]. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by [[John William Strutt, 3rd Baron Rayleigh|Lord Rayleigh]], [[Walther Ritz]], and [[Boris Galerkin]].

The finiteapplication elementof methodFEM obtainedgained its real impetusmomentum in the 1960s and 1970s bydue to the developments of [[John Argyris|J. H. Argyris]] withand his co-workers at the [[University of Stuttgart]],; [[Ray W. Clough|R. W. Clough]] withand his co-workers at [[University of California, Berkeley|UCUniversity of California Berkeley]],; [[Olgierd Zienkiewicz|O. C. Zienkiewicz]] withand his co-workers [[Ernest Hinton]], [[Bruce Irons (engineer)|Bruce Irons]],<ref>{{Cite journal |last1=Hinton |first1=Ernest |last2=Irons |first2=Bruce |title=Least squares smoothing of experimental data using finite elements |journal=Strain |volume=4 |issue=3 |pages=24–27 |date=July 1968 |doi= 10.1111/j.1475-1305.1968.tb01368.x}}</ref> and others at [[Swansea University]],; [[Philippe G. Ciarlet]] at the University of [[Pierre-and-Marie-Curie University|Paris 6]]; and [[Richard H. Gallagher|Richard Gallagher]] withand his co-workers at [[Cornell University]]. FurtherDuring impetusthis wasperiod, providedadditional inimpetus thesewas yearsprovided by the available open-source finite elementFEM programs. NASA sponsored the original version of [[NASTRAN]]. UCUniversity of California Berkeley made the finite element programs SAP IV<ref>{{cite web |title=SAP-IV Software and Manuals |url=http://nisee.berkeley.edu/elibrary/getpkg?id=SAP4 |___location=NISEE e-Library, The Earthquake Engineering Online Archive |access-date=2013-01-24 |archive-date=2013-03-09 |archive-url=https://web.archive.org/web/20130309013628/http://nisee.berkeley.edu/elibrary/getpkg?id=SAP4 |url-status=live }}</ref> and, later, [[OpenSees]] widely available. In Norway, the ship classification society Det Norske Veritas (now [[DNV GL]]) developed [[SESAM (FEM)|Sesam]] in 1969 for use in the analysis of ships.<ref>{{cite book |title=Building Trust, The history of DNV 1864-2014 |author1=Gard Paulsen |author2=Håkon With Andersen |author3=John Petter Collett |author4=Iver Tangen Stensrud |date=2014 |publisher=Dinamo Forlag A/S |isbn=978-82-8071-256-1 |___location=Lysaker, Norway |pages=121, 436}}<!-- |access-date=30 June 2015 --></ref> A rigorous mathematical basis tofor the finite element methodFEM was provided in 1973 with thea publication by [[Gilbert Strang]] and [[George Fix]].<ref>{{cite book |first1=Gilbert |last1=Strang |author-link1=Gilbert Strang |first2=George |last2=Fix |author-link2=George Fix |title=An Analysis of The Finite Element Method |url=https://archive.org/details/analysisoffinite0000stra |url-access=registration |publisher=Prentice Hall |year=1973 |isbn=978-0-13-032946-2}}</ref> The method has since been generalized for the [[numerical analysis|numerical modeling]] of physical systems in a wide variety of [[engineering]] disciplines, e.g.,such as [[electromagnetism]], [[heat transfer]], and [[fluid dynamics]].<ref name="ZienkiewiczTaylor2013">{{cite book |author1=Olek C Zienkiewicz |author2=Robert L Taylor |author3=J.Z. Zhu |title=The Finite Element Method: Its Basis and Fundamentals |url=https://books.google.com/books?id=7UL5Ls9hOF8C |date=31 August 2013 |publisher=Butterworth-Heinemann |isbn=978-0-08-095135-5}}</ref><ref>{{cite book |first1=K.J. |last1=Bathe |author-link1= Klaus-Jürgen Bathe |title=Finite Element Procedures |publisher= Cambridge, MA: Klaus-Jürgen Bathe |year=2006 |isbn= 978-0979004902}}</ref>

==Technical discussion==

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==Discretization==

[[File: Finite element method 1D illustration1.pngsvg|thumb|A function in <math>H_0^1,</math> with zero values at the endpoints (blue) and a piecewise linear approximation (red)]]

P1 and P2 are ready to be discretized, which leads to a common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem:

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===Generalized finite element method===

The generalized finite element method (GFEM) uses local spaces consisting of functions, not necessarily polynomials, that reflect the available information on the unknown solution and thus ensure good local approximation. Then a [[partition of unity]] is used to “bond”"bond" these spaces together to form the approximating subspace. The effectiveness of GFEM has been shown when applied to problems with domains having complicated boundaries, problems with micro-scales, and problems with boundary layers.<ref>{{cite journal | first1=Ivo | last1=Babuška | author1-link=Ivo Babuška | first2=Uday | last2=Banerjee | first3=John E. | last3=Osborn | author3-link=John E. Osborn (mathematician) | title=Generalized Finite Element Methods: Main Ideas, Results, and Perspective | journal=[[International Journal of Computational Methods]] | date=June 2004 | issue=1 | pages= 67–103 | doi=10.1142/S0219876204000083 | volume=1}}</ref>

===Mixed finite element method===

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The [[finite difference method]] (FDM) is an alternative way of approximating solutions of PDEs. The differences between FEM and FDM are:

* The most attractive feature of the FEM is its ability to handle complicated geometries (and boundaries) with relative ease. While FDM in its basic form is restricted to handle rectangular shapes and simple alterations thereof, the handling of geometries in FEM is theoretically straightforward.<ref name=":0" /><ref name=":1">{{Cite journal |last=Topper |first=Jürgen |date=January 2005 |title=Option pricing with finite elements |url=http://dx.doi.org/10.1002/wilm.42820050119 |journal=Wilmott |volume=2005 |issue=1 |pages=84–90 |doi=10.1002/wilm.42820050119 |doi-broken-date=2024-04-07 |issn=1540-6962|url-access=subscription }}</ref>

* FDM is not usually used for irregular CAD geometries but more often for rectangular or block-shaped models.<ref>{{Cite news|url=http://www.machinedesign.com/fea-and-simulation/what-s-difference-between-fem-fdm-and-fvm|title=What's The Difference Between FEM, FDM, and FVM?|date=2016-04-18|work=Machine Design|access-date=2017-07-28|archive-date=2017-07-28|archive-url=https://web.archive.org/web/20170728024918/http://www.machinedesign.com/fea-and-simulation/what-s-difference-between-fem-fdm-and-fvm|url-status=live}}</ref>

* FEM generally allows for more flexible mesh adaptivity than FDM.<ref name=":1" />

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== Finite element and fast fourier transform (FFT) methods ==

Another method used for approximating solutions to a partial differential equation is the [[Fast Fourier transform|Fast Fourier Transform]] (FFT), where the solution is approximated by a fourier series computed using the FFT. For approximating the mechanical response of materials under stress, FFT is often much faster,<ref>{{Cite journal |last1=Ma |first1=X |last2=Parvathaneni |first2=K |last3=Lomov |first3=S |last4=Vasiukov |first4=D |last5=Shakoor |first5=M |last6=Park |first6=C |title=Quantitative comparison between fast fourier transform and finite element method for micromechanical modeling of composite |url=https://hal.science/hal-02416258 |journal=FiBreMoD Conference|date=December 2019 }}</ref> but FEM may be more accurate.<ref name=":2">{{Cite journal |last1=Prakash |first1=A |last2=Lebensohn |first2=R A |date=2009-09-01 |title=Simulation of micromechanical behavior of polycrystals: finite elements versus fast Fourier transforms |url=https://iopscience.iop.org/article/10.1088/0965-0393/17/6/064010 |journal=Modelling and Simulation in Materials Science and Engineering |volume=17 |issue=6 |pages=064010 |doi=10.1088/0965-0393/17/6/064010 |bibcode=2009MSMSE..17f4010P |issn=0965-0393|url-access=subscription }}</ref> One example of the respective advantages of the two methods is in simulation of [[Rolling (metalworking)|rolling]] a sheet of [[Aluminium|aluminum]] (an FCC metal), and [[Wire drawing|drawing]] a wire of [[tungsten]] (a BCC metal). This simulation did not have a sophisticated shape update algorithm for the FFT method. In both cases, the FFT method was more than 10 times as fast as FEM, but in the wire drawing simulation, where there were large deformations in [[Crystallite|grains]], the FEM method was much more accurate. In the sheet rolling simulation, the results of the two methods were similar.<ref name=":2" /> FFT has a larger speed advantage in cases where the boundary conditions are given in the materials [[Strain (mechanics)|strain]], and loses some of its efficiency in cases where the [[Stress (mechanics)|stress]] is used to apply the boundary conditions, as more iterations of the method are needed.<ref>{{Cite journal |last1=Cruzado |first1=A |last2=Segurado |first2=J |last3=Hartl |first3=D J |last4=Benzerga |first4=A A |date=2021-06-01 |title=A variational fast Fourier transform method for phase-transforming materials |url=https://iopscience.iop.org/article/10.1088/1361-651X/abe4c7 |journal=Modelling and Simulation in Materials Science and Engineering |volume=29 |issue=4 |pages=045001 |doi=10.1088/1361-651X/abe4c7 |bibcode=2021MSMSE..29d5001C |issn=0965-0393|url-access=subscription }}</ref>

The FE and FFT methods can also be combined in a [[voxel]] based method (2) to simulate deformation in materials, where the FE method is used for the macroscale stress and deformation, and the FFT method is used on the microscale to deal with the effects of microscale on the mechanical response.<ref name=":3">{{Cite journal |last1=Gierden |first1=Christian |last2=Kochmann |first2=Julian |last3=Waimann |first3=Johanna |last4=Svendsen |first4=Bob |last5=Reese |first5=Stefanie |date=2022-10-01 |title=A Review of FE-FFT-Based Two-Scale Methods for Computational Modeling of Microstructure Evolution and Macroscopic Material Behavior |journal=Archives of Computational Methods in Engineering |language=en |volume=29 |issue=6 |pages=4115–4135 |doi=10.1007/s11831-022-09735-6 |issn=1886-1784|doi-access=free }}</ref> Unlike FEM, FFT methods’ similarities to image processing methods means that an actual image of the microstructure from a microscope can be input to the solver to get a more accurate stress response. Using a real image with FFT avoids meshing the microstructure, which would be required if using FEM simulation of the microstructure, and might be difficult. Because fourier approximations are inherently periodic, FFT can only be used in cases of periodic microstructure, but this is common in real materials.<ref name=":3" /> FFT can also be combined with FEM methods by using fourier components as the variational basis for approximating the fields inside an element, which can take advantage of the speed of FFT based solvers.<ref>{{Cite journal |last1=Zeman |first1=J. |last2=de Geus |first2=T. W. J. |last3=Vondřejc |first3=J. |last4=Peerlings |first4=R. H. J. |last5=Geers |first5=M. G. D. |date=2017-09-07 |title=A finite element perspective on nonlinear FFT-based micromechanical simulations: A FINITE ELEMENT PERSPECTIVE ON NONLINEAR FFT-BASED SIMULATIONS |url=https://onlinelibrary.wiley.com/doi/10.1002/nme.5481 |journal=International Journal for Numerical Methods in Engineering |language=en |volume=111 |issue=10 |pages=903–926 |doi=10.1002/nme.5481|arxiv=1601.05970 }}</ref>

==Application==

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Various specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in the design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and minimizing weight, materials, and costs.<ref name="Engineering Asset Management">{{cite journal|last1=Kiritsis |first1=D. |last2=Eemmanouilidis |first2=Ch. |last3=Koronios |first3=A. |last4=Mathew |first4=J. |date=2009 |title=Engineering Asset Management |journal=Proceedings of the 4th World Congress on Engineering Asset Management (WCEAM) |pages=591–592}}</ref>

FEM allows detailed visualization of where structures bend or twist, indicating the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of modeling and system analysis. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured. The mesh is an integral part of the model and must be controlled carefully to give the best results. Generally, the higher the number of elements in a mesh, the more accurate the solution of the discretized problem. However, there is a value at which the results converge, and further mesh refinement does not increase accuracy.<ref>{{Cite web |url=https://coventivecomposites.com/explainers/finite-element-analysis-how-to-create-a-great-model/ |title=Finite Element Analysis: How to create a great model |date=2019-03-18 |website=Coventive Composites |language=en-GB |access-date=2019-04-05 }}{{Dead link|date=May 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

[[File:Human knee joint FE model.png|thumb|245x245px|Finite Element Model of a human knee joint<ref>{{Cite journal| last1=Naghibi Beidokhti| first1=Hamid| last2=Janssen| first2=Dennis| last3=Khoshgoftar| first3=Mehdi| last4=Sprengers| first4=Andre| last5=Perdahcioglu| first5=Emin Semih| last6=Boogaard| first6=Ton Van den| last7=Verdonschot| first7=Nico| title=A comparison between dynamic implicit and explicit finite element simulations of the native knee joint| journal=Medical Engineering & Physics| volume=38| issue=10| pages=1123–1130| doi=10.1016/j.medengphy.2016.06.001| pmid=27349493| year=2016| url=https://ris.utwente.nl/ws/files/6153316/CMBBE2014-Hamid-Submitted.pdf| access-date=2019-09-19| archive-date=2018-07-19| archive-url=https://web.archive.org/web/20180719212657/https://ris.utwente.nl/ws/files/6153316/CMBBE2014-Hamid-Submitted.pdf| url-status=live}}</ref>]]

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In the 1990s FEM was proposed for use in stochastic modeling for numerically solving probability models<ref>{{cite journal |title=Methods with high accuracy for finite element probability computing |author1=Peng Long |author2=Wang Jinliang |author3=Zhu Qiding |journal=Journal of Computational and Applied Mathematics |volume=59 |issue=2 |date=19 May 1995 |pages=181–189 |doi=10.1016/0377-0427(94)00027-X |doi-access= }}</ref> and later for reliability assessment.<ref>{{cite book |first1=Achintya |last1=Haldar |first2=Sankaran |last2=Mahadevan |title=Reliability Assessment Using Stochastic Finite Element Analysis |publisher=John Wiley & Sons |isbn=978-0471369615 |year=2000}}</ref>

FEM is widely applied for approximating differential equations that describe physical systems. This method is very popular in the community of [[Computational fluid dynamics]], and there are many applications for solving [[Navier–Stokes equations]] with FEM.<ref>{{cite book |last1=Girault |first1=Vivette |last2=Raviart |first2=Pierre-Arnaud |title=Finite Element Approximation of the Navier-Stokes Equations |volume=749 |year=1979 |publisher=Springer Berlin|isbn=978-3-540-09557-6}}</ref><ref>{{cite book |last1=Cuvelier |first1=Cornelis |last2=Segal |first2=August |last3=Van Steenhoven |first3=Anton A |title=Finite Element Methods and Navier-Stokes Equations |volume=22 |year=1986 |publisher=Springer Science & Business Media|isbn=978-1-4020-0309-7}}</ref><ref>{{cite book |last1=Girault |first1=Vivette |last2=Raviart |first2=Pierre-Arnaud |title=Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms |volume=5 |year=2012 |publisher=Springer Science & Business Media|isbn=978-3-642-64888-5}}</ref> Recently, the application of FEM has been increasing in the researches of computational plasma. Promising numerical results using FEM for [[Magnetohydrodynamics]], [[Vlasov equation]], and [[Schrödinger equation]] have been proposed.<ref>{{cite journal |last1=Karakashian |first1=Ohannes |last2=Makridakis |first2=Charalambos |year=1999 |title=A Space-Time Finite Element Method for the Nonlinear Schrödinger Equation: The Continuous Galerkin Method |journal=SIAM Journal on Numerical Analysis |publisher=SIAM |volume=36 |issue=6 |pages=1779–1807 |doi=10.10161137/0377-0427(94)00027-XS0036142997330111 |doi-access= |year=1999 |publisher=SIAM}}</ref><ref>{{cite journal |last1=Sovinec |first1=Carl R. |last2=Glasser |first2=A.H. |last3=Gianakon |first3=T.A. |last4=Barnes |first4=D.C. |last5=Nebel |first5=R.A. |last6=Kruger |first6=S.E. |last7=Schnack |first7=D.D. |last8=Plimpton |first8=S.J. |last9=Tarditi |first9=A. |last10=Chu |first10=M.S. |title=Nonlinear Magnetohydrodynamics Simulation Using High-Order Finite Elements |journal=Journal of Computational Physics |volume=195 |issue=1 |pages=355–386 |year=2004 |doi=10.1016/j.jcp.2003.10.004 |publisher=Elsevier|bibcode=2004JCoPh.195..355S }}</ref>

==See also==

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*[[Patch test (finite elements)|Patch test]]

*[[Rayleigh–Ritz method]]

*[[SDC Verifier]]

*[[Space mapping]]

*[[STRAND7]]