Finite Element Method - a universal way of solving differential equations

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In modern science, there are many approaches to build a quantitative mathematical model of any system.And one of them is considered to be the finite element method, which is based on the establishment of the behavior of the differential (infinitesimal) of its elements, based on the alleged relationship between the basic elements that are able to give a full description of this system.Thus, this technique is used to describe a system of differential equations.

Theoretical aspects

Theoretical methods headed by the finite difference method, which is the ancestor of the series of tools for calculation and is widely used.The finite difference method is especially attractive is their application to any differential equations.However, because of the cumbersome and difficult programmability account boundary conditions for the problem, there are some limitations in the application of these techniques.The accuracy of the solution depends on the level of the grid, which defines the key points.Therefore, in solving problems of this type often have to consider the system of algebraic equations of higher order.

finite element method - an approach that has reached a very high level of accuracy.Today, many scientists point out that at present there is no similar method that can give the same results.The finite element method has a wide range of applicability, efficiency and ease with which takes into account the actual boundary conditions, allowed to become a serious contender for any other method.However, beyond these advantages, it is characterized by some drawbacks.For example, it contains the sampling circuit, which inevitably entails use of a large number of elements.Especially when it comes to three-dimensional problems, which have removed the borders and within each of them all the unknown variables can be traced continuity.

alternative approach

Alternatively, some scientists propose the use of analytical integration of differential equations by other means or by introducing some approximations.In any case, no matter what method is used, first of all must be integrated differential equation.As the first stage of solving the problem is necessary to transform the differential equations in the integral analogues.This operation allows you to get a system of equations having a value within a specific area.

Another alternative approach is the boundary element method, the development of which is built on the idea of ‚Äč‚Äčintegral equations.This method is widely used without evidence of the uniqueness of each individual decision, so it is becoming very popular and is being implemented with the use of computer technology.

Scope

finite element method quite successfully used in conjunction with other numerical methods in a mixed formulation.This combination allows you to expand the scope of its application.