В современной науке существует множество подходов 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 determination of the behavior of the differential (infinitely small) of its element, based on the assumed relationship between the basic elements that are able to give a full characteristic of this system. Thus, this technique uses differential equations in the description of the system.
Theoretical aspects
Theoretical methods heads the finite methoddifferences, which is the ancestor of this series of calculus tools and is widely used. In finite difference methods, their particular application to any differential equations is a particular attraction. However, due to the bulkiness and difficult programmability of taking into account the boundary conditions in 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 determines the nodal points. Therefore, when solving problems of this type, it is often necessary to consider systems of algebraic equations of a higher order.
The finite element method is an approach that has reachedvery high level of accuracy. And today, many scientists note that at the present stage there is no similar method that can give the same results. The finite element method has a wide range of applicability, its effectiveness and the ease with which the actual boundary conditions are taken into account make it possible to become a serious contender for any other method. However, in addition to these advantages, it has some drawbacks. For example, it is represented by a sampling scheme, which inevitably entails the use of a large number of elements. Especially if we are talking about three-dimensional problems that have remote boundaries, and within each of them continuity is traced across all unknown variables.
Alternative approach
As an alternative to some scientistsIt is proposed to use analytical integration of a system of differential equations in some other way or by introducing some approximation. In any case, no matter what method is used, the differential equation must first be integrated. As a first step in solving the problem, it is necessary to convert differential equations into a system of integral analogs. This operation allows you to obtain a system of equations that has values within a specific area.
Another alternative approach is the methodboundary elements whose development is based on the idea of integral equations. This method is widely used without proof of uniqueness in each individual solution, so that it becomes very popular and is implemented using computer technology.
Application area
The finite element method is quite successfully used in combination with other numerical methods in mixed formulation. This combination allows you to expand its scope.
