Slovník pojmů

Finite-element method

Finite-element method is a general numerical method of solving partial differential equations. The method consists of the following steps:

1. An analyzed structure is subdivided into sub-regions (finite elements), which do not overlap on one hand, and which contain all the points of the analyzed structure on the other hand. In the region of a finite element, parameters of the analyzed structure (permittivity, permeability, conductivity) have to be constant. There are no restrictions with respect to the size and to the shape of finite elements.

2. If the analyzed structure is subdivided into finite elements, an unknown function (solution) can be approximated in formal way over the whole finite element. The unknown solution is usually approximated by linear combination of known, suitable elected basis functions and unknown approximation coefficients. That way, a single equation for M unknown approximation coefficients is obtained. If approximation coefficients are computed, an approximation of unknown quantity over the whole analyzed structure is obtained.

3. If the approximation is substituted to the solved equation, the equation is not met completely due to the approximate nature of the approximation. This fact is respected by introducing a residual function. If the approximation is accurate then the residual function is small. Therefore, the residual function is going to be minimized. The minimization is performed by the method of weighted residua.

4. Applying the method of weighted residua, the residual function is multiplied by a proper weighting function, the product is integrated over the whole analyzed space, and the result is put to equal zero. Exploiting as much properly elected weighting functions as much unknown approximation coefficients is contained in the formal approximation, a set of simultaneous linear equations is obtained, which solution uniquely determines approximation coefficients.

5. Computing approximation coefficients, a solution of the problem is obtained. Substituting approximation coefficients to the formal approximation, an approximation of the solution is obtained in every point of every finite element. Integrating approximations over finite elements, a global approximation of the solution is obtained for every point of the analyzed structure.