In modern science, there are many approaches to constructing 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 basic elements that can give a complete characterization of this system. Thus, this technique uses differential equations when describing the system.
Theoretical aspects
Theoretical methods are headed by the method of finite differences, which is the ancestor of this series of calculus tools and is widely used. In finite difference methods, their application to any differential equations is particularly attractive . However, due to the cumbersomeness and difficult programmability of accounting for boundary conditions, there are some limitations in the application of these methods. The accuracy of the solution depends on the level of the grid, which defines the nodal points. Therefore, in solving problems of this type, it is often necessary to consider systems of higher-order algebraic equations.
The finite element method is an approach that has reached a very high level of accuracy. And today, many scientists note that at the present stage there is no analogous method capable of producing the same results. The finite element method has a wide range of applicability, its efficiency and 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 is characterized by 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 for all unknown variables.
Alternative approach
As an alternative, some scientists propose the use of analytic integration of a system of differential equations in another way, or by introducing some approximation. In any case, whatever method is used, the differential equation must first be integrated. As the first stage of solving the problem, it is necessary to transform the differential equations into a system of integral analogs. This operation allows us to obtain a system of equations that has values within a particular region.
Another alternative approach is the method of boundary elements, the development of which is based on the idea of integral equations. This method is widely used without evidence of uniqueness in each individual solution, making it very popular and implemented using computer technology.
Application area
The finite element method is used quite successfully in combination with other numerical methods in mixed formulation. This combination allows us to expand the scope of its application.