A finite element method (abbreviated as FEM) is a numerical technique to obtain an approximate solution to a class of problems governed by elliptic partial differential equations. Such problems are called as boundary value problems as they consist of a partial differential equation and the boundary conditions. The finite element method converts the elliptic partial differential equation into a set of algebraic equations which are easy to solve. The initial value problems which consist of a parabolic or hyperbolic differential equation and the initial conditions (besides the boundary conditions) can not be completely solved by the finite element method. The parabolic or hyperbolic differential equations contain the time as one of the independent variables. To convert the time or temporal derivatives into algebraic expressions, another numerical technique like the finite difference method (FDM) is required. Thus, to solve an initial value problem, one needs both the finite element method as well as the finite difference method where the spatial derivatives are converted into algebraic expressions by FEM and the temporal derivatives are converted into algebraic equations by FDM.
In mathematics the finite element method FEM is a numerical technique for finding approximate solutions to boundary value problems for differential equations It uses variational methods the calculus of variations to minimize an error function and produce a stable solution Analogous to the idea that connecting many tiny straight lines can approximate a larger circle FEM 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.
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