94 Applied calculus of variations for engineers
0
0
.2
0
.4
0
.6
0
.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1
y_approx(x)
y_exact(x)
y_error(x)
FIGURE 7.1 Accuracy of the Ritz solution
The figure demonstrates that the Ritz solution satisfies the boundary con-
ditions and shows acceptable differences in the interior of the interval. Finally,
the variational problem’s extremum is computed for both cases. The analyt-
ical solution is based on the derivative
y
=
√
2πcos(πx),
and obtained as
1
0
y
2
(x)dx =2π
2
1
0
cos
2
(πx)dx = π
2
=9.87.
The Ritz solution’s derivative is
y
= −
√
30(2x − 1).
and the approximate extremum is
1
0
y
2
(x)dx =30
1
0
(2x − 1)
2
dx =
30
3
=10.
The approximate extremum is slightly higher than the analytic extremum,
but by only a very acceptable error.
Numerical methods of calculus of variations 95
7.2.1 Application: solution of Poisson’s equation
The second order boundary value problem of Poisson’s, introduced earlier, is
presented in the variational form of
I(y)=
D
((
∂u
∂x
)
2
+(
∂u
∂y
)
2
+2f (x, y)u(x, y))dxdy
whose Euler-Lagrange equation leads to the form
∂
2
u
∂x
2
+
∂
2
u
∂y
2
= f(x, y),
as was shown in the last chapter. For the simplicity of the discussion and
without the loss of generality, we impose the boundary condition of
u =0
on the perimeter of the domain D. Ritz’s method indicates the use of the
basis functions
b
i
(x, y)
and demands that they also vanish on the boundary. The approximate solu-
tion in this two-dimensional case is
u(x, y)=
n
i=1
α
i
b
i
(x, y).
The partial derivatives are
∂
u
∂x
=
n
i=1
α
i
∂b
i
(x, y)
∂x
,
and
∂
u
∂y
=
n
i=1
α
i
∂b
i
(x, y)
∂y
.
Substituting the approximate solution into the functional yields
I(
u)=
D
((
∂
u
∂x
)
2
+(
∂
u
∂y
)
2
+2f (x, y)u(x, y))dxdy.
Evaluating the derivatives, this becomes
I(
u)=
D
((
n
i=1
α
i
∂b
i
∂x
)
2
+(
n
i=1
α
i
∂b
i
∂y
)
2
+2f (x, y))
n
i=1
α
i
b
i
)dxdy,
which may be reordered into the form
I(
u)=
n
i=1
n
j=1
c
ij
α
i
α
j
+
n
i=1
d
i
α
i
.
96 Applied calculus of variations for engineers
The coefficients are
c
ij
=
D
(
∂b
i
∂x
∂b
j
∂x
+
∂b
i
∂y
∂b
j
∂y
)dxdy
and
d
i
=
D
f(x, y)b
i
dxdy.
As above, the unknown coefficients are solved from the conditions
∂I(
u)
∂α
i
=0,i=1, 2,...,n,
resulting in the linear system of equations
n
j=1
c
ij
α
j
+ d
j
=0,i=1, 2,...,n.
It may be shown that the system is nonsingular and always yields a nontrivial
solution assuming that the basis functions form a linearly independent set.
The computation of the terms of the equations, however, is rather tedious
and resulted in the emergence of the next method.
7.3 Galerkin’s method
The difference between the Ritz method and that of Galerkin’s is in the fact
that the latter addresses the differential equation form of a variational prob-
lem. Galerkin’s method minimizes the residual of the differential equation
integrated over the domain with a weight function; hence, it is also called the
method of weighted residuals.
This difference lends more generality and computational convenience to
Galerkin’s method. Let us consider a linear differential equation in two vari-
ables:
L(u(x, y)) = 0
and apply Dirichlet boundary conditions. Galerkin’s method is also based on
the Ritz approximation of the solution as
u =
n
i=1
α
i
b
i
(x, y),
Numerical methods of calculus of variations 97
in which case, of course there is a residual of the differential equation
L(
u) =0.
Galerkin proposed using the basis functions of the approximate solution also
as the weights, and requires that the integral vanishes with a proper selection
of the coefficients:
D
L(u)b
j
(x, y)dxdy =0;j =1, 2,...,n.
This yields a system for the solution of the coefficients as
D
L(
n
i=1
α
i
b
i
(x, y))b
j
(x, y)dxdy =0;j =1, 2,...,n.
This is also a linear system and produces the unknown coefficients α
i
. We il-
lustrate the computational technique of Galerkin’s method also in connection
with Poisson’s equation:
L(u)=
∂
2
u
∂x
2
+
∂
2
u
∂y
2
− f(x, y)=0.
For this Galerkin’s method is presented as
D
(
∂
2
u
∂x
2
+
∂
2
u
∂y
2
− f(x, y))b
j
dxdy =0,j =1,...,n.
Therefore
D
(
n
i=1
α
i
∂
2
b
i
∂x
2
+
n
i=1
α
i
∂
2
b
i
∂y
2
− f(x, y))b
j
dxdy =0,j =1,...,n.
Reordering yields
n
i=1
α
i
D
(
∂
2
b
i
∂x
2
+
∂
2
b
i
∂y
2
)b
j
dxdy −
D
f(x, y)b
j
dxdy =0,j =1,...,n.
The system of equations becomes
Ba
= b
with solution vector of
a
=
⎡
⎢
⎢
⎣
α
1
α
2
...
α
n
⎤
⎥
⎥
⎦
.
98 Applied calculus of variations for engineers
The system matrix is of the form
B =
⎡
⎢
⎢
⎣
B
1,1
B
1,2
... B
1,n
B
2,1
B
2,2
... B
2,n
... ... ... ...
B
n,1
B
n,2
... B
n,n
⎤
⎥
⎥
⎦
whose terms are defined as
B
j,i
=
D
(
∂
2
b
i
∂x
2
+
∂
2
b
i
∂y
2
)b
j
dxdy.
Finally, the right-hand side vector is
b
=
⎡
⎢
⎢
⎣
D
f(x, y)b
1
dxdy
D
f(x, y)b
2
dxdy
...
D
f(x, y)b
n
dxdy
⎤
⎥
⎥
⎦
.
Note that if Galerkin’s and Ritz’s methods are applied to the same problem,
the approximate solutions found are identical.
7.4 Kantorovich’s method
Both the Ritz and Galerkin methods are restricted in their choices of basis
functions, because their basis functions are required to satisfy the boundary
conditions. The clever method of Kantorovich, described in [13], relaxes this
restriction and enables the use of simpler basis functions.
Consider the variational problem of two variables
I(u)=extremum, (x, y) ∈ D,
with boundary conditions
u(x, y)=0, (x, y) ∈ ∂D.
Here ∂D again denotes the boundary of the domain.
The method proposes the construction of a function, such that
ω(x, y) ≥ 0, (x, y) ∈ D,
and
ω(x, y)=0, (x, y) ∈ ∂D.
Get Applied Calculus of Variations for Engineers, 2nd Edition now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
