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    Finite Element Method

    Finite element method is a numerical technique for finding approximate solutions to boundary value problems. It uses variation methods to minimize an error function and produce a stale solution. Finite element method encompasses all the methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain.

    A typical work out of the method involves dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem, followed by systematically recombining all sets of element equations into a global system of equations for the final calculation.

    The element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often partial differential equations. The finite element method is commonly introduced as a special case of Galerkin method. The process is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero.

    Finite element methods are a good choice for analyzing problems over complicated domains, when the domain changes, when the desired precision varies over the entire domain, or when the solution lacks smoothness. An example of this would be in numerical weather prediction where it is more important to have accurate predictions over developing highly nonlinear phenomena rather than relatively calm areas. 

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