The method of Galerkin's weighted residuals [Sch80][Zie89][Oña91] is used as a general formulation for the discretization of partial differential equations even if an extreme formulation can not be found as for example in case of Ritz's approach [Sch80][Sch97b] .
In Galerkin's approach the basic equation is weighted with appropriate discrete weighting functions and the resulting equation integrated over the region of interest is equaled to zero.
For the discretization of the Laplace operator
= = 0 | (3.1) |
To solve (3.2) a spatial variation of
must be
found. The first step is to subdivide the simulation domain into a
number of subdomains called elements (Fig. 3.1). Each
element is associated with a number of discrete points or nodes
located within the element or on its boundary. The spatial variation of
within an element is then defined in terms of the values at the
belonging node points. If the element has n node points with
associated values
, , ... ,
the value of
can be formulated as
Having completed a spatial subdivision of the simulation domain and
after allocation of suitable shape functions, the next step is to
integrate (3.2) over each element. This is usually
accomplished by some form of numerical integration (see also
Chapter 3.1.2) leading to an element contribution of
the form
h^{ . } | (3.4) |
All element contributions are summed up and the resulting
final system matrix H