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===Choosing a basis===
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To complete the discretization, we must select a [[Basis (linear algebra)|basis]] of <math>V</math>. In the one-dimensional case, for each control point <math>x_k</math> we will choose the piecewise linear function <math>v_k</math> in <math>V</math> whose value is <math>1</math> at <math>x_k</math> and zero at every <math>x_j,\;j \neq k</math>, i.e.,
<math display="block">v_{k}(x) = \begin{cases}
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[[File:Finite element triangulation.svg|thumb|Solving the two-dimensional problem <math>u_{xx}+u_{yy}=-4</math> in the disk centered at the origin and radius 1, with zero boundary conditions.<br />(a) The triangulation.]]
[[File:Finite element sparse matrix.png|thumb|(b) The [[sparse matrix]] ''L'' of the discretized linear system]]
[[File:Finite element solution.svg|thumb|(c) The computed solution, <math>u(x, y)=1-x^2-y^2
The primary advantage of this choice of basis is that the inner products
<math display="block">\langle v_j,v_k \rangle = \int_0^1 v_j v_k\,dx</math>
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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
This powerful design tool has significantly improved both the standard of engineering designs and the design process methodology in many industrial applications.<ref name=Hastings>Hastings, J. K., Juds, M. A., Brauer, J. R., ''Accuracy and Economy of Finite Element Magnetic Analysis'', 33rd Annual National Relay Conference, April 1985.</ref> The introduction of FEM has substantially decreased the time to take products from concept to the production line.<ref name=Hastings/> Testing and development have been accelerated primarily through improved initial prototype designs using FEM.<ref name="McLaren-Mercedes">{{cite web|title=McLaren Mercedes: Feature - Stress to impress |author=McLaren-Mercedes |year=2006 |url=http://www.mclaren.com/features/technical/stress_to_impress.php |access-date=2006-10-03 |archive-url=https://web.archive.org/web/20061030200423/http://www.mclaren.com/features/technical/stress_to_impress.php |archive-date=2006-10-30 |url-status=dead }}</ref> In summary, benefits of FEM include increased accuracy, enhanced design and better insight into critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive design cycle, increased productivity, and increased revenue.<ref name=Hastings/>
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