The finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains. Consider the following model problem:
\[
−
Δ
u
=
f
in
Ω
C
,
u
=
g
on
∂
Ω
,
∂
u
∂
r
+
1
r
u
=
o
(
1
r
)
as
r
=
|
x
|
→
∞
,
- \Delta u = f\quad {\text {in}}\;{\Omega ^C},\quad u = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial u}}{{\partial r}} + \frac {1}{r}u = o\left ( {\frac {1}{r}} \right )\quad {\text {as}}\;r = |x| \to \infty ,
\]
where
Ω
C
{\Omega ^C}
is the complement in
R
3
{R^3}
(three-dimensional Euclidean space) of a bounded set
Ω
\Omega
with smooth boundary
∂
Ω
\partial \Omega
, f and g are smooth functions, and f has bounded support. This problem is approximately solved by introducing an artificial boundary
Γ
R
{\Gamma _R}
near infinity, e.g. a sphere of sufficiently large radius R. The intersection of this sphere with
Ω
C
{\Omega ^C}
is denoted by
Ω
R
C
\Omega _R^C
and the given problem is replaced by
\[
−
Δ
u
R
=
f
in
Ω
R
C
,
u
R
=
g
on
∂
Ω
,
∂
u
R
∂
r
+
1
r
u
R
=
0
on
Γ
R
.
- \Delta {u_R} = f\quad {\text {in}}\;\Omega _R^C,\quad {u_R} = g\quad {\text {on}}\;\partial \Omega ,\quad \frac {{\partial {u_R}}}{{\partial r}} + \frac {1}{r}{u_R} = 0\quad {\text {on}}\;{\Gamma _R}.
\]
This problem is then solved approximately by the finite element method, resulting in an approximate solution
u
R
h
u_R^h
for each
h
>
0
h > 0
. In order to obtain a reasonably small error for
u
−
u
R
h
=
(
u
−
u
R
)
+
(
u
R
−
u
R
h
)
u - u_R^h = (u - {u_R}) + ({u_R} - u_R^h)
, it is necessary to make R large. This necessitates the solution of a large number of linear equations, so that this method is often not very good when a uniform mesh size h is employed. It is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by
C
h
−
3
C{h^{ - 3}}
with C independent of h and R.