
Recent developments of meshless methods 83
3.3.3 Examples for validation
In this subsection, several 1-D and 2-D examples of ordinary and partial
differential equations that are always associated with mechanics problems
are solved to demonstrate the performance of the PWLS method. The fol-
lowing norms are dened as the error indicators,
=
∑
=
=
e
N
u
1| |
i
N
i
N
i
0
1
exact num
1
exact
(3.90)
where u
i
exact
and u
i
num
are the exact and numerical approximate solutions
of the function, respectively. The errors for the rst-order derivatives of the
function are dened as e
1x
and e
1y
, respectively
=
∑−
∑
=
∑
=
=
=
=
e
N
uu
u
e
N
u
1| |
||
,
1
x
i
N
ix ix
i
N
ix
y
i
N
iy iy ...