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| publisher = SIAM
| doi = 10.1137/030601077
| hdl = 20.500.11850/147656
| hdl-access = free
}}</ref> To solve a problem, 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 that 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
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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 smaller elements, as well as the use of software coded with a FEM algorithm. When 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 PDEs or [[integral equation]]s, while the divided, smaller elements of the complex problem represent different areas in the physical system.
FEA may be used for analyzing problems over complicated domains (e.g., cars and oil pipelines) when the ___domain changes (e.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>{{Cite journal |date=2022 |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 }}</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 the 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>
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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
===Mixed finite element method===
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