The finite element method has become a well established method in many fields of computer aided engineering, such as structural analysis, fluid dynamics, and electromagnetic field computation. However, its flexibility in modeling arbitrary geometries comes at the cost of a more complex mathematical background.
There are tree main steps during the solution of a partial differential equation (PDE) with the finite element method. First, the domain, on which the PDE should be solved, is discretized into finite elements. Depending on the dimension of the problem these can be triangles, squares, or rectangles in two dimensions or tetrahedrons, cubes, or hexahedra for three dimensional problems. The solution of the PDE is approximated by piecewise continuous polynomials and the PDE is hereby discretized and split into a finite number of algebraic equations. Thus, the aim is to determine the unknown coefficients of these polynomials in such a way, that the distance (which is defined by the norm in a suitable vector space) from the exact solution becomes a minimum. Therefore, the finite element method is essentially a minimization technique for variational problems. Since the number of elements is finite, we have reduced the problem of finding a continuous solution for our PDE to calculating the finite number of coefficients of the polynomials.