<s>
The	O
patch	B-Algorithm
test	I-Algorithm
in	O
the	O
finite	B-Application
element	I-Application
method	I-Application
is	O
a	O
simple	O
indicator	O
of	O
the	O
quality	O
of	O
a	O
finite	B-Application
element	I-Application
,	O
developed	O
by	O
Bruce	O
Irons	O
.	O
</s>
<s>
The	O
patch	B-Algorithm
test	I-Algorithm
uses	O
a	O
partial	O
differential	O
equation	O
on	O
a	O
domain	O
consisting	O
from	O
several	O
elements	O
set	O
up	O
so	O
that	O
the	O
exact	O
solution	O
is	O
known	O
and	O
can	O
be	O
reproduced	O
,	O
in	O
principle	O
,	O
with	O
zero	O
error	O
.	O
</s>
<s>
The	O
elements	O
pass	O
the	O
patch	B-Algorithm
test	I-Algorithm
if	O
the	O
finite	B-Application
element	I-Application
solution	O
is	O
the	O
same	O
as	O
the	O
exact	O
solution	O
.	O
</s>
<s>
It	O
was	O
long	O
conjectured	O
by	O
engineers	O
that	O
passing	O
the	O
patch	B-Algorithm
test	I-Algorithm
is	O
sufficient	O
for	O
the	O
convergence	O
of	O
the	O
finite	B-Application
element	I-Application
,	O
that	O
is	O
,	O
to	O
ensure	O
that	O
the	O
solutions	O
from	O
the	O
finite	B-Application
element	I-Application
method	I-Application
converge	O
to	O
the	O
exact	O
solution	O
of	O
the	O
partial	O
differential	O
equation	O
as	O
the	O
finite	B-Application
element	I-Application
mesh	O
is	O
refined	O
.	O
</s>
<s>
However	O
,	O
this	O
is	O
not	O
the	O
case	O
,	O
and	O
the	O
patch	B-Algorithm
test	I-Algorithm
is	O
neither	O
sufficient	O
nor	O
necessary	O
for	O
convergence	O
.	O
</s>
<s>
A	O
broader	O
definition	O
of	O
patch	B-Algorithm
test	I-Algorithm
(	O
applicable	O
to	O
any	O
numerical	O
method	O
,	O
including	O
and	O
beyond	O
finite	B-Application
elements	I-Application
)	O
is	O
any	O
test	O
problem	O
having	O
an	O
exact	O
solution	O
that	O
can	O
,	O
in	O
principle	O
,	O
be	O
exactly	O
reproduced	O
by	O
the	O
numerical	O
approximation	O
.	O
</s>
<s>
Therefore	O
,	O
a	O
finite-element	O
simulation	O
that	O
uses	O
linear	O
shape	O
functions	O
has	O
patch	B-Algorithm
tests	I-Algorithm
for	O
which	O
the	O
exact	O
solution	O
must	O
be	O
piecewise	O
linear	O
,	O
while	O
higher-order	O
finite	B-Application
elements	I-Application
have	O
correspondingly	O
higher-order	O
patch	B-Algorithm
tests	I-Algorithm
.	O
</s>
