<s>
In	O
mathematics	O
,	O
in	O
the	O
area	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
Galerkin	B-Algorithm
methods	I-Algorithm
are	O
named	O
after	O
the	O
Soviet	O
mathematician	O
Boris	O
Galerkin	O
.	O
</s>
<s>
They	O
convert	O
a	O
continuous	O
operator	O
problem	O
,	O
such	O
as	O
a	O
differential	O
equation	O
,	O
commonly	O
in	O
a	O
weak	B-Algorithm
formulation	I-Algorithm
,	O
to	O
a	O
discrete	O
problem	O
by	O
applying	O
linear	O
constraints	O
determined	O
by	O
finite	O
sets	O
of	O
basis	O
functions	O
.	O
</s>
<s>
Often	O
when	O
referring	O
to	O
a	O
Galerkin	B-Algorithm
method	I-Algorithm
,	O
one	O
also	O
gives	O
the	O
name	O
along	O
with	O
typical	O
assumptions	O
and	O
approximation	O
methods	O
used	O
:	O
</s>
<s>
Ritz	B-Algorithm
–	I-Algorithm
Galerkin	I-Algorithm
method	I-Algorithm
(	O
after	O
Walther	O
Ritz	O
)	O
typically	O
assumes	O
symmetric	B-Algorithm
and	O
positive	O
definite	O
bilinear	O
form	O
in	O
the	O
weak	B-Algorithm
formulation	I-Algorithm
,	O
where	O
the	O
differential	O
equation	O
for	O
a	O
physical	O
system	O
can	O
be	O
formulated	O
via	O
minimization	O
of	O
a	O
quadratic	O
function	O
representing	O
the	O
system	O
energy	O
and	O
the	O
approximate	O
solution	O
is	O
a	O
linear	O
combination	O
of	O
the	O
given	O
set	O
of	O
the	O
basis	O
functions	O
.	O
</s>
<s>
Bubnov	B-Algorithm
–	I-Algorithm
Galerkin	I-Algorithm
method	I-Algorithm
(	O
after	O
Ivan	O
Bubnov	O
)	O
does	O
not	O
require	O
the	O
bilinear	O
form	O
to	O
be	O
symmetric	B-Algorithm
and	O
substitutes	O
the	O
energy	O
minimization	O
with	O
orthogonality	B-Application
constraints	O
determined	O
by	O
the	O
same	O
basis	O
functions	O
that	O
are	O
used	O
to	O
approximate	O
the	O
solution	O
.	O
</s>
<s>
In	O
an	O
operator	O
formulation	O
of	O
the	O
differential	O
equation	O
,	O
Bubnov	B-Algorithm
–	I-Algorithm
Galerkin	I-Algorithm
method	I-Algorithm
can	O
be	O
viewed	O
as	O
applying	O
an	O
orthogonal	O
projection	O
to	O
the	O
operator	O
.	O
</s>
<s>
Petrov	O
–	O
Galerkin	B-Algorithm
method	I-Algorithm
(	O
after	O
Georgii	O
I	O
.	O
Petrov	O
)	O
allows	O
using	O
basis	O
functions	O
for	O
orthogonality	B-Application
constraints	O
(	O
called	O
test	O
basis	O
functions	O
)	O
that	O
are	O
different	O
from	O
the	O
basis	O
functions	O
used	O
to	O
approximate	O
the	O
solution	O
.	O
</s>
<s>
Petrov	O
–	O
Galerkin	B-Algorithm
method	I-Algorithm
can	O
be	O
viewed	O
as	O
an	O
extension	O
of	O
Bubnov	B-Algorithm
–	I-Algorithm
Galerkin	I-Algorithm
method	I-Algorithm
,	O
applying	O
a	O
projection	O
that	O
is	O
not	O
necessarily	O
orthogonal	O
in	O
the	O
operator	O
formulation	O
of	O
the	O
differential	O
equation	O
.	O
</s>
<s>
Examples	O
of	O
Galerkin	B-Algorithm
methods	I-Algorithm
are	O
:	O
</s>
<s>
the	O
Galerkin	B-Algorithm
method	I-Algorithm
of	O
weighted	O
residuals	O
,	O
the	O
most	O
common	O
method	O
of	O
calculating	O
the	O
global	O
stiffness	B-Algorithm
matrix	I-Algorithm
in	O
the	O
finite	B-Application
element	I-Application
method	I-Application
,	O
</s>
<s>
the	O
boundary	B-Algorithm
element	I-Algorithm
method	I-Algorithm
for	O
solving	O
integral	O
equations	O
,	O
</s>
<s>
Krylov	B-Algorithm
subspace	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Let	O
us	O
introduce	O
Galerkin	B-Algorithm
's	I-Algorithm
method	I-Algorithm
with	O
an	O
abstract	O
problem	O
posed	O
as	O
a	O
weak	B-Algorithm
formulation	I-Algorithm
on	O
a	O
Hilbert	O
space	O
,	O
namely	O
,	O
</s>
<s>
Reducing	O
the	O
problem	O
to	O
a	O
finite-dimensional	O
vector	O
subspace	O
allows	O
us	O
to	O
numerically	B-General_Concept
compute	O
as	O
a	O
finite	O
linear	O
combination	O
of	O
the	O
basis	O
vectors	O
in	O
.	O
</s>
<s>
Subtracting	O
the	O
two	O
,	O
we	O
get	O
the	O
Galerkin	O
orthogonality	B-Application
relation	O
for	O
the	O
error	O
,	O
which	O
is	O
the	O
error	O
between	O
the	O
solution	O
of	O
the	O
original	O
problem	O
,	O
,	O
and	O
the	O
solution	O
of	O
the	O
Galerkin	O
equation	O
,	O
</s>
<s>
Since	O
the	O
aim	O
of	O
Galerkin	B-Algorithm
's	I-Algorithm
method	I-Algorithm
is	O
the	O
production	O
of	O
a	O
linear	O
system	O
of	O
equations	O
,	O
we	O
build	O
its	O
matrix	O
form	O
,	O
which	O
can	O
be	O
used	O
to	O
compute	O
the	O
solution	O
algorithmically	O
.	O
</s>
<s>
Due	O
to	O
the	O
definition	O
of	O
the	O
matrix	O
entries	O
,	O
the	O
matrix	O
of	O
the	O
Galerkin	O
equation	O
is	O
symmetric	B-Algorithm
if	O
and	O
only	O
if	O
the	O
bilinear	O
form	O
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
While	O
this	O
is	O
not	O
really	O
a	O
restriction	O
of	O
Galerkin	B-Algorithm
methods	I-Algorithm
,	O
the	O
application	O
of	O
the	O
standard	O
theory	O
becomes	O
much	O
simpler	O
.	O
</s>
<s>
Furthermore	O
,	O
a	O
Petrov	O
–	O
Galerkin	B-Algorithm
method	I-Algorithm
may	O
be	O
required	O
in	O
the	O
nonsymmetric	O
case	O
.	O
</s>
<s>
First	O
,	O
we	O
will	O
show	O
that	O
the	O
Galerkin	O
equation	O
is	O
a	O
well-posed	B-Algorithm
problem	I-Algorithm
in	O
the	O
sense	O
of	O
Hadamard	O
and	O
therefore	O
admits	O
a	O
unique	O
solution	O
.	O
</s>
<s>
By	O
the	O
Lax-Milgram	B-Algorithm
theorem	I-Algorithm
(	O
see	O
weak	B-Algorithm
formulation	I-Algorithm
)	O
,	O
these	O
two	O
conditions	O
imply	O
well-posedness	B-Algorithm
of	O
the	O
original	O
problem	O
in	O
weak	B-Algorithm
formulation	I-Algorithm
.	O
</s>
<s>
Therefore	O
,	O
the	O
well-posedness	B-Algorithm
of	O
the	O
Galerkin	O
problem	O
is	O
actually	O
inherited	O
from	O
the	O
well-posedness	B-Algorithm
of	O
the	O
original	O
problem	O
.	O
</s>
<s>
Since	O
the	O
proof	O
is	O
very	O
simple	O
and	O
the	O
basic	O
principle	O
behind	O
all	O
Galerkin	B-Algorithm
methods	I-Algorithm
,	O
we	O
include	O
it	O
here	O
:	O
</s>
<s>
by	O
ellipticity	O
and	O
boundedness	O
of	O
the	O
bilinear	O
form	O
(	O
inequalities	O
)	O
and	O
Galerkin	O
orthogonality	B-Application
(	O
equals	O
sign	O
in	O
the	O
middle	O
)	O
,	O
we	O
have	O
for	O
arbitrary	O
:	O
</s>
<s>
For	O
simplicity	O
of	O
presentation	O
in	O
the	O
section	O
above	O
we	O
have	O
assumed	O
that	O
the	O
bilinear	O
form	O
is	O
symmetric	B-Algorithm
and	O
positive	O
definite	O
,	O
which	O
implies	O
that	O
it	O
is	O
a	O
scalar	O
product	O
and	O
the	O
expression	O
is	O
actually	O
a	O
valid	O
vector	O
norm	O
,	O
called	O
the	O
energy	O
norm	O
.	O
</s>
<s>
Dividing	O
by	O
and	O
taking	O
the	O
infimum	O
over	O
all	O
possible	O
proves	O
that	O
the	O
Galerkin	B-Algorithm
approximation	I-Algorithm
is	O
the	O
best	O
approximation	O
in	O
the	O
energy	O
norm	O
within	O
the	O
subspace	O
,	O
i.e.	O
</s>
<s>
studied	O
the	O
application	O
of	O
the	O
Galerkin	B-Algorithm
method	I-Algorithm
to	O
stepped	O
structures	O
.	O
</s>
<s>
Gander	O
and	O
Wanner	O
showed	O
how	O
Ritz	O
and	O
Galerkin	B-Algorithm
methods	I-Algorithm
led	O
to	O
the	O
modern	O
finite	B-Application
element	I-Application
method	I-Application
.	O
</s>
