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4.1.1 Galerkin's Method
Multiplying (4.1) by a function , which is called test or trial function, and integrating over the simulation domain gives the variational formulation

(4.2) 
Using the notation

(4.3) 
(4.2) can be written as

(4.4) 
In order to obtain the corresponding discrete problem, the simulation domain, , is divided in a set of elements,
, which do not overlap, i.e.
. The mesh obtained by such a domain discretization is represented by

(4.5) 
Further, one defines a set of grid points, also called nodes, with each point being described by a unique global index
, where is the total number of grid points in the mesh.
The approximate solution,
, for the unknown function, , is given by [152]

(4.6) 
where
are the socalled basis (or shape) functions. The approximate solution of (4.4) is determined by the coefficients , which represent the value of the unknown function at the node .
At the node , where the point is given by the coordinates , the basis functions must satisfy the condition

(4.7) 
Typically, the basis functions are chosen to be low order polynomials.
Substituting (4.6) in (4.4), and choosing
one obtains

(4.8) 
and since is a linear operator and the coefficients are constants one can write

(4.9) 
Equation (4.9) is, in fact, a linear system of equations with unknowns,
. Thus, it can be written in matrix notation as

(4.10) 
where
is called stiffness matrix, given by the elements

(4.11) 
is the vector of unknown coefficients, and
is the load vector, given by

(4.12) 
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R. L. de Orio: Electromigration Modeling and Simulation