4.1.2 Assembly

Applying the finite element method to solve a given PDE leads to an algebraic system of equations. In order to solve this system of equations, the global stiffness matrix, $\mathbf{A}$, and the load vector, $\mathbf{b}$, have to be determined. However, instead of computing them using directly (4.11) and (4.12), in practice they are computed by summing the contributions from the different elements [152,153,154] according to

$$a_{ij} = \sum_{T\in T_h(\symDomain)} \left(L[N_i], N_j\right)_T =... ...\symDomain)} \int_{T} L[N_i(\vec r)] N_j(\vec r) d\symDomain,\qquad i,j=1,...,N$$ (4.13)

$$b_{j} = \sum_{T\in T_h(\symDomain)} \left(f, N_j\right)_T = \sum_... ... T_h(\symDomain)} \int_{T} f(\vec r) N_j(\vec r) d\symDomain, \qquad j=1,...,N.$$ (4.14)

Note that $\left(L[N_i], N_j\right)_T=0$ unless both $N_i$ and $N_j$ belong to the same element $\T$. Thus, the calculations (4.13) and (4.14) can be limited to the nodes of the element $\T$, so that $i,j = 1,..., N_V$, where $N_V$ is the number of vertices of the element. In this way, for each element $T\in T_h(\symDomain)$, a $N_V\times N_V$ matrix is obtained, which is called element stiffness or nucleus matrix. Thus, the general system matrix, $\mathbf{A}$, can be computed by first computing the nucleus matrices for each $T\in T_h(\symDomain)$ and then summing the contributions from each element according to (4.13) [152]. The right-hand side vector, $\mathbf{b}$, is computed in the same way. This process of constructing the general system matrix is called assembly [152]. The main advantage of this assembly process is that it greatly simplifies the computation of the system matrix and right-hand side vector, since (4.11) and (4.12) can be easily calculated for each element of the domain discretization.