SKEDSOFT

Note :   The nodes in the mesh are numbered in an orderly way. We number them from left to right and from top to bottom or from bottom to top. A typical numbering is given in figure 1.2 (a) and  (b)

                              Figure 1.2 (a) Numbering of nodes.                                                                     Figure 1.2 (b) : Numbering of nodes.

System of equations governing the solutions:

The difference approximation 

or

to the Laplace equation uxx uyy = ∇2u = 0 is applied at all the nodes and the boundary conditions are used to simplify the equations. The resulting system is a linear system of algebraic equations Au = d.

Structure of the coefficient matrix :

Let us write the system of equations that arise when
we have nine nodes as given in Fig.1.2 (a) . Since the boundary values are known, we have the following system of equations.

where b₁, b₂, b₃, b₄, b₆, b₇, b₈, b₉ are the contributions from the boundary values.