Finite Element Method (FEM)

Finite Element Method (FEM) The FE method is similar to the FV method in many ways. The domain is broken  into a set of  discrete volumes or finite elements that are generally unstructured; in 2D, they are usually triangles or quadrilaterals, while in 3D  tetrahedra or hexahedra are most often used. The distinguishing feature of FE methods is that the equations are multiplied by a weight function before they are integrated over the entire domain. In the simplest FE methods, the solution is approximated by a linear shape function within each element in a way that guarantees continuity of  the solution across  element boundaries. Such a function can be constructed from  its values at the corners of  the elements. The weight function is usually of the same  form.         
 This approximation  is  then  substituted  into  the  weighted  integral  of  the  conservation law and the equations to be solved are derived  by requiring the derivative  of  the integral with respect to each nodal value to be  zero; this corresponds to selecting the best solution within the set of allowed functions (the one with minimum  residual). The result  is a set of non-linear algebraic equations.       
  An important  advantage  of  finite  element methods is the ability  to deal with arbitrary geometries. Finite element methods are relatively easy to analyze mathematically and can be shown to have optimality properties for certain  types of equations. The principal drawback, which is shared by any method that uses unstructured grids, is that the matrices of the linearized equations are not as well structured as those for regular grids making it more difficult to find efficient solution methods. 

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