7.4 Potential Equation 417
Figure 4 Mesh numbering forL-shaped region.tem of equations to be solved will be rather less regular than that for a rectan-
gle.
Example.
Consider the problem
∂^2 u
∂x^2 +∂^2 u
∂y^2 =−^16 inR, (15)
u= 0 on the boundary ofR, (16)whereRis anL-shaped region formed from a 1×1squarebyremovinga1/ 4 ×
1 /4 square from the upper right corner. The general replacement equation is
uN+uS+uE+uW− 4 ui=− 1. (17)
With the numbering shown in Fig. 4, the eight equations to be solved are
u 2 +u 4 − 4 u 1 =− 1
u 1 +u 3 +u 5 − 4 u 2 =− 1
u 2 +u 6 − 4 u 3 =− 1
u 1 +u 5 +u 7 − 4 u 4 =− 1
u 2 +u 4 +u 6 +u 8 − 4 u 5 =− 1
u 3 +u 5 − 4 u 6 =− 1
u 4 +u 8 − 4 u 7 =− 1
u 5 +u 7 − 4 u 8 =− 1.(18)
The results, rounded to three digits, are shown in Eq. (19). Note the equalities,
which arise from symmetries in the problem:
u 1 = 0. 656
u 2 =u 4 = 0. 813
u 3 =u 7 = 0. 616
u 5 = 0. 981
u 6 =u 8 = 0. 649.(19)