Next: 4.1.3 Shape Function Up: 4.1 The Finite Element Previous: 4.1.1 Galerkin's Method

## 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, , and the load vector, , 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

 (4.13)

 (4.14)

Note that unless both and belong to the same element . Thus, the calculations (4.13) and (4.14) can be limited to the nodes of the element , so that , where is the number of vertices of the element. In this way, for each element , a matrix is obtained, which is called element stiffness or nucleus matrix. Thus, the general system matrix, , can be computed by first computing the nucleus matrices for each and then summing the contributions from each element according to (4.13) [152]. The right-hand side vector, , 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.

Next: 4.1.3 Shape Function Up: 4.1 The Finite Element Previous: 4.1.1 Galerkin's Method

R. L. de Orio: Electromigration Modeling and Simulation