The finite element method with nonuniform mesh sizes for unbounded domains
Authors: Goldstein C. I. 
Published: June 2010

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:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus normal upper Delta u equals f in normal upper Omega Superscript upper C Baseline comma u equals g on partial-differential normal upper Omega comma StartFraction partial-differential u Over partial-differential r EndFraction plus StartFraction 1 Over r EndFraction u equals o left-parenthesis StartFraction 1 Over r EndFraction right-parenthesis as r equals StartAbsoluteValue x EndAbsoluteValue right-arrow normal infinity comma">

−
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus normal upper Delta u Subscript upper R Baseline equals f in normal upper Omega Subscript upper R Superscript upper C Baseline comma u Subscript upper R Baseline equals g on partial-differential normal upper Omega comma StartFraction partial-differential u Subscript upper R Baseline Over partial-differential r EndFraction plus StartFraction 1 Over r EndFraction u Subscript upper R Baseline equals 0 on normal upper Gamma Subscript upper R Baseline period">

−
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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u minus u Subscript upper R Superscript h Baseline equals left-parenthesis u minus u Subscript upper R Baseline right-parenthesis plus left-parenthesis u Subscript upper R Baseline minus u Subscript upper R Superscript h Baseline right-parenthesis">

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.