Editing Computational fluid dynamics (section)

==== Finite element method ====

{{Main|Finite element method}}

The finite element method (FEM) is used in structural analysis of solids, but is also applicable to fluids.  However, the FEM formulation requires special care to ensure a conservative solution. The FEM formulation has been adapted for use with fluid dynamics governing equations.<ref>{{Cite web |title=Detailed Explanation of the Finite Element Method (FEM) |url=https://www.comsol.com/multiphysics/finite-element-method |access-date=2022-07-15 |website=www.comsol.com}}</ref><ref name=":0">{{Cite book |last=Anderson |first=John David |url=https://books.google.com/books?id=phG_QgAACAAJ |title=Computational Fluid Dynamics: The Basics with Applications |date=1995 |publisher=McGraw-Hill |isbn=978-0-07-113210-7 |language=en}}</ref> Although FEM must be carefully formulated to be conservative, it is much more stable than the finite volume approach.<ref>{{cite journal| title=k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions| last1=Surana| first1=K.A.| last2=Allu| first2=S.| last3=Tenpas| first3=P.W.| last4=Reddy| first4=J.N.| journal=International Journal for Numerical Methods in Engineering| volume=69| issue=6| pages=1109–1157|date=February 2007| doi=10.1002/nme.1801|bibcode = 2007IJNME..69.1109S | s2cid=122551159}}</ref> FEM also provides more accurate solutions for smooth problems comparing to FVM. <ref>{{cite journal |last1=Surana |first1=KS |last2=Allu |first2=S |last3=Tenpas |first3=PW |last4=Reddy |first4=JN |title=k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions |journal=International Journal for Numerical Methods in Engineering |volume=69 |issue=6 |pages=1109–1157 |year=2007 |publisher=Wiley Online Library|doi=10.1002/nme.1801 |bibcode=2007IJNME..69.1109S }}</ref> Another advantage of FEM is that it can handle complex geometries and boundary conditions. However, FEM can require more memory and has slower solution times than the FVM.<ref>{{cite journal |last1=Surana |first1=KS |last2=Allu |first2=S |last3=Tenpas |first3=PW |last4=Reddy |first4=JN |title=k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions |journal=International Journal for Numerical Methods in Engineering |volume=69 |issue=6 |pages=1109–1157 |year=2007 |publisher=Wiley Online Library|doi=10.1002/nme.1801 |bibcode=2007IJNME..69.1109S }}</ref>

In this method, a weighted residual equation is formed:

:<math>R_i = \iiint W_i Q \, dV^e</math>

where <math>R_i</math> is the equation residual at an element vertex <math>i</math>, <math>Q</math> is the conservation equation expressed on an element basis, <math>W_i</math> is the weight factor, and <math>V^{e}</math> is the volume of the element.