In modern science, there are many approaches to building 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 element, based on the assumed relationship between the main elements that can give a complete description of this system. Thus, this technique uses differential equations to describe the system.
Theoretical aspects
Theoretical methods are headed by the finite difference method, which is the founder of this series of calculus tools and is widely used. In finite difference methods, their application to any
differential equations is especially attractive
. However, due to the cumbersomeness and difficult programmability of accounting for 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 achieved a very 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 ease with which the actual boundary conditions are taken into account allow us to become a serious competitor for any other method. However, in addition to these advantages, it also has some disadvantages. For example, it is represented by a discretization scheme, which inevitably entails the use of a large number of elements. Especially when it comes to three-dimensional problems that have remote boundaries, and within each of them, continuity is traced across all unknown variables.
Alternative approach
As an alternative, some scientists suggest using analytical integration of the system of differential equations in another way or by introducing some approximation. In any case, no matter what method is used, the differential equation must first be integrated. As the first step in solving the problem, it is necessary to transform the differential equations into a system of integral analogues. This operation allows you to get a system of equations having values within a specific area.
Another alternative approach is the boundary element method, the development of which is based on the idea of integral equations. This method is widely used without proof of uniqueness in each individual solution, so it becomes very popular and is implemented using computer technology.
Application area
The finite element method is rather successfully used in combination with other numerical methods in mixed formulations. This combination allows you to expand the scope of its application.