4.2.6 Solution of the Linear Equations System

The nodal values can be determined by solving the system of equations (4.26). Once they are known, they can be substituted in the finite element approximation φ_hⁱ within each element (equation (4.3)) in order to determine an appropriate approximate solution that best satisfies the original DE in the domain of interest. The linear system of equations (4.26) has only two unknowns and can easily be solved. In general, matrices have thousands of unknowns and finding the solution of such a system of linear equations constitutes the most computationally demanding part of FEM. In general, a linear system of N equations with N unknowns, c_n, can be written in matrix notation [148] as

\[\begin{equation} [K]\{c\}=\{f\}+\{\beta\}=\{L\}, \end{equation}\] (4.27)

where {L} is the load vector. The solution of the global matrix equations computes the unknowns as follows

\[\begin{equation} \{c\}=[K]^{-1}\{L\}. \end{equation}\] (4.28)

There are two fundamental classes of algorithms which are employed to solve equation (4.28), namely iterative method and direct method. An iterative method, such as the Jacobi method, generates a sequence of approximate solutions of the problem and uses a given initial approximation to generate the successive ones. It therefore approaches the solution gradually until the sequence of approximations converges for the given initial approximate solution. A direct method, such as Gaussian elimination, provides the exact solution of the problem by a finite number of operations. It typically uses more memory than the iterative solvers.