Applied Mathematics, 2011, 2, 1327-1338
doi:10.4236/am.2011.211186 Published Online November 2011 (http://www.SciRP.org/journal/am)
An Introduction to Numerical Methods for the Solutions of
Partial Differential Equations
Manoj Kumar, Garima Mishra
Department of Mat hematics, Motilal Nehru National Institute of Technology, Allahabad, India
Received August 8, 2011; revised September 27, 2011; accepted October 5, 2011
Abstract
Partial differential equations arise in formulations of problems involving functions of several variables such
as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity, etc. The pre-
sent paper deals with a general introduction and classification of partial differential equations and the nu-
merical methods available in the literature for the solution of partial differential equations.
Keywords: Partial Differential Equations, Eigenvalue, Finite Difference Method, Finite Volume Method,
Finite Element Method
1. Introduction
An equation involving derivatives or differentials of one
or more dependent variables with respect to one or more
independent variables is called a differential equation.
The study of differential equations is a wide field in pure
and applied mathematics, physics, meteorology, and en-
gineering, etc. All of these disciplines are concerned with
the properties of differential equations of various types.
Pure mathematics focuses on the existence and uni-
queness of solutions, while applied mathematics empha-
sizes the rigorous justification of the methods for appro-
ximating solutions. Differential equations play an im-
portant role in modeling virtually every physical, tech-
nical, or biological process, from celestial motion, to bri-
dge design, and interactions between neurons. Diffe-
rential equations which are used to solve real-life pro-
blems may not necessarily be directly solvable, that is,
do not have closed form solutions. Instead, solutions can
be approximated using numerical methods. Mathema-
ticians also study weak solutions (relying on weak de-
rivatives), which are types of solutions that do not have
to be differentiable everywhere. This extension is often
necessary for solutions to exist, and it also results in
more physically reasonable properties of solutions, such
as possible presence of shocks for equations of hyper-
bolic type.
The theory of differential equations is quite developed
and the methods used to study them vary significantly
with the type of the equation.
A differential equation involving derivatives with re-
spect to single independent variable is called an ordinary
differential equation. In the simplest form, the dependent
variable is a real or complex valued function, but more
generally, it may be vector-valued or matrix-valued: this
corresponds to considering a system of ordinary differen-
tial equations for a single variable. Ordinary differential
equations are classified according to the order of the
highest derivative of the dependent variable with respect
to the independent variable appearing in the equation.
The most important cases for applications are first-order
and second-order differential equations. In the classical
literature, the distinction is also made between diffe-
rential equations explicitly solved with respect to the
highest derivative and differential equations in an im-
plicit form.
A differential equation involving partial derivatives
with respect to two or more independent variables is
called partial differential equation. The partial differen-
tial equations can also be classified on basis of highest
order derivative.
Some topics in differential geometry as minimal sur-
faces and imbedding problems, which give rise to the
Monge-Ampere equations, have stimulated the analysis
of partial differential equations, especially nonlinear equ-
ations. Moreover, the theory of systems of first order
partial differential equations has a significant interaction
with Lie theory and with the work of E. Cartan.
The development of partial differential equations in
the 18th and 19th century is given in Kline’s book [1].

M. KUMAR ET AL. 1328
Until the 1870, the study of partial differential equation
was mainly concerned with heuristic methods for finding
solutions of boundary value problems as well as explicit
solutions for particular problems (for an example, the
solution of Dirichlet boundary value problem =0, u

introduced by Riemann).
2
in ,u
Poincaré [2] gave the first complete proof of the exi-
stence and uniqueness of a solution of the Laplace equa-
tion for any continuous Dirichlet boundary condition in
1890. In a fundamental paper of, Poincare [3] established
the existence of an infinite sequence of eigenvalues and
corresponding eigen-functions for the Laplace operator
under the Dirichlet boundary condition. Picard applied
the method of successive approximation to obtain solu-
tions of nonlinear problems which were mild perturb-
ations of uniquely solvable linear problems. The constru-
ction of elementary solutions and Green’s functions for
general higher order linear elliptic operators was carried
through to the analytic case by E. E. Levi [4]. Up to
about 1920’s solutions of partial differential equations
were generally understood to be classical solutions, that
is, for a differential operator of order
k
C.k
Keeping in view the requirement of the new resear-
chers, the present paper describes the basic fundamentals
of partial differential equations which has been collected
from a large number of research articles published in re-
puted journals and literature available in the books with
the intension to provide all important relevant material in
a condense form related to partial differential equations
and numerical methods for their solutions. Also, since
analytical and computational solution of partial diffe-
rential equations is the major concern from the early
years, this paper gives a small step towards the deve-
lopment of computational analysis of partial differential
equations, which have lot of utilization in the field of
science and engineering.
2. Classification of Partial Differential
Equations
Both ordinary and partial differential equations are broa-
dly classified as linear and nonlinear. A linear partial di-
fferential equation is one in which all of the partial deri-
vatives appears in linear form and none of the coeffi-
cients depends on the dependent variables. The coeffici-
ent may be function of the independent variables. A non-
linear partial differential equation can be described as a
partial differential equation involving nonlinear terms.
2.1. Types of Non-Linear Partial Differential
Equations
The non-linear partial differential equations describe many
different physical systems, ranging from gravitation to
fluid dynamics and have been used in mathematics to
solve problems such as Poincare conjecture and Calabi
conjecture.
A non-linear partial differential equation is known as
semi-linear if it is linear in highest order derivatives and
the coefficients of highest order derivatives depend only
on independent variables.


1
0
||=
,,,, =0
k
k
aDuaDu Duux






. (1)
A nonlinear partial differential equation is known as
quasi-linear if it is linear in highest order derivatives and
the coefficients of highest order derivatives depend on
independent variables as well on lesser order derivatives.



1
||=
1
0
,, ,,
,,,, =0
k
k
k
aDu DuuxDu
aDu Duux







.
(2)
A nonlinear partial differential equation is known as
fully non-linear if the coefficients depends on dependent
variable or the derivatives appear in nonlinear form.
Example 2.1
= is linear equation
xx yy x y
fffffxy

  (3)
2= is semilinear
xxyy xy
afbfff c (4)
2= 0 is quasilinear
xxxyyy x
ffffff (5)
= 0 is nonlinear
xx yyxy
fffaf bf

 (6)
where a, b are functions of x, y and c is function of x, y
and f.
But further classification into elliptic, hyperbolic, and
parabolic equations, especially for second-order linear
equations, is of utmost importance. For more study on
linear and quasi linear elliptic equations, see [5,6].
2.2. Classification Based on Discriminant
The general quasi linear second-order non-homogeneous
partial differential equation in two independent variable
is
=
xx xyyyxy
A
fBfCfDfEfFfG

 (7)
The classification of partial differential equations de-
pends on the sign of discriminant as fallow:
24BAC
,
,
,
1) If the partial differential equation
is hyperbolic.
24>0BAC
2) If the partial differential equation
is parabolic.
24=0BAC
3) If the partial differential equation
is elliptic.
24<0BAC

M. KUMAR ET AL.1329
n



2.3. Eigenvalue Based Classification of Partial
Differential Equations
Since method for classification of partial differential equ-
ations fails if it is partial differential equation in three or
more independent variable then we are not able to find
discriminant.
Let us consider a general second order partial equation
in n independent variables
11 12131
11 12131
21 22232
21 22232
12 3
123
12 3
123 =0.
xxxxxxn xx
n
xxxxxxn xx
n
nxxn xxnxxnnxx
nnn n
xxx nx
n
au auauau
au auauau
auauauau
bubububu cu









(8)
which also can be written in compact form as:
2
=1 =1=1
= 0.
nn n
ij i
ij i
ij i
uu
abcu
xx x



  (9)
Then coefficient matrix of highest order derivatives is
=[ ]
ij
A
a.
Let

be an eigenvalue of
A
corresponding to ei-
genvector
X
=
A
XX

=0AX X


()=AIX0


= 0 Since X0AI  
then
11 121
21 222
12
=0










n
n
nn nn
aa a
aa a
aa a
.
The characterisic equation will have roots as n
eigenvalues.
n
Characterization of differential equation is based on
following:
 If any eigenvalue is zero, then partial differential equ-
ation is parabolic.
 If all eigenvalues are non-zero and one eigenvalue
has opposite sign, then partial differential equation is
hyperbolic.
 If all eigenvalues are nonzero having same sign, then
partial differential equation is elliptic.
Example 2.2 Consider the following differential equ-
ation of flow:

22
2
22
1=Mxy





where
M

is Mac number.
1) First we characterize by discriminant
2
22
=0, =1, =1,
then 4=4(1) = 4(1).
BAMC
BACMM



2

 
 If flow is subsonic i.e. <1,M then
24<BAC0
0
0
.
The equation of flow is elliptic.
 If flow is sonic i.e. =1,M then
24=BAC.
The equation of flow is parabolic.
 If flow is subsonic i.e. >1,M then
24>BAC.
The equation of flow is hyperbolic.
2) Now by eigenvalues
2
10
=01
M
A









2
2
10
==
01



1,1




M
AI M
.
 If <1,M then all eigenvalues are nonzero and of
same sign thus equation of flow is elliptic.
 If =1,M then all eigenvalues are zero thus equ-
ation of flow is parabolic.
 If >1,M then all eigenvalues are nonzero and of
one eigenvalue of opposite sign thus equation of flow
is hyperbolic.
2.4. Significance of Classification
The classification of a partial differential equations is
intimately related to the characteristics of the partial
differential equations. The characteristics are


1n

-
dimensional hyper-surfaces in n-dimensional hyper-
space that have some very special features. In two-dim-
ensional space, which is the case considered generally,
characteristics are paths in the solution domain along
which information propagates. In other words we can say
information propagates throughout the solution domain
along characteristics.
2.5. Classification by Physical Problems
Physical problems falls into one of the following general
classification:
1) Equilibrium Problem;
2) Propagation Problem;
3) Eigenvalue Problem.
Equilibrium Problem: Equilibrium problems are
steady state problems in closed domain in
which the solution
(, )Dxy
(, )
f
xy is governed by an elliptic
0,

M. KUMAR ET AL. 1330
partial differential equations subject to boundary con-
ditions specified at each point on the boundary of
the domain.
B
).
Such type of problems have no real characteristic and
thus the solution at every point in the solution domain is
influenced by the solution at all other points and the
solution at each points influence the solution at all other
points.
Equilibrium problems are solved by method of relaxa-
tion numerically.
Propagation Problem: Propagation problems are ini-
tial value problems in open domains. Here by open do-
main means that open with respect to one of the in-
dependent variables.
Example 2.3
0
= with initial condition (,) =(
txx
f
ffxtfx

Eigenvalue Problem: Eigenvalue problem are special
problems in which the solution exits only for special
values (i.e. eigenvalues) of a parameter of the problem.
The eigenvalues are to be determined in addition to the
corresponding configuration of the system.
2.6. Types of Conditions
Initial Value Problem: An initial value problem is one
in which the dependent variable and possibly its deriva-
tives are specified initially (i.e. at time ) or at the
same value of independent variable in the equation.
Initial value problems are generally time-dependent pro-
blems.
=0t
Example 2.4 2
0
2
=, (,)=(
uu
uxtf x
tx


).
Boundary Value Problems: A boundary value pro-
blem is one in which the dependent variable and possibly
its derivatives are specified at the extreme of the in-
dependent variable. For steady state equilibrium pro-
blems, the auxiliary conditions consists of boundary con-
ditions on the entire boundary of the closed solution
domain. There are three types of boundary condition.
Example 2.5 Let be a bounded domain in
with a smooth boundary and let
n 
, :f 
be a locally Hölder continuous function. The BVP
=(,),in ufxu (10)
=0,on u

 (11)
is called nonlinear elliptic boundary value problem. This
type of BVP arises in several domains, for example in
physical problems involving the steady-state tempera-
ture distribution see [7-9].
1) Dirichlet boundary condition: The value of the
function is specified on the boundary. The depen-
dent variable of the partial differential equation are pre-
scribed in domain at different points. For example if an
iron rod had one end held at absolute zero then the value
of the problem would be known at that point in space. A
Dirichlet boundary condition imposed on an ordinary or
a partial differential equation specifies the values of a
solution is to take on the boundary of the domain. The
question of finding solutions to such equations is known
as the Dirichlet problem as in (10).
f
2) Neumann boundary condition: The value of de-
rivative normal to the boundary is specified (
f
n


is
specified on the boundary). For example if one iron rod
had heater at one end then energy would be added at a
constant rate but the actual temperature would not be
known. A Neumann boundary condition imposed on an
ordinary or a partial differential equation specifies the
derivative values of a solution is to take on the boundary
of the domain. For example
=(,),in ufxu


=0,on .
u
x


3) Mixed boundary condition: The linear combi-
nation of Dirichlet and Neumann boundary conditions:
f
afb n


is specified on the boundary.
Mixed boundary conditions are also known as Cauchy
boundary condition. A Cauchy boundary condition im-
posed on an ordinary or a partial differential equation
specifies both the values a solution of a differential equ-
ation is to take on the boundary of the domain and the
normal derivative at the boundary.
=(,),in ufxu


12
=0,on .
u
ccu
x




3. Various Methods for Solving Partial
Differential Equation
In literature various method exits for solution of partial
differential equations. Here we will discuss some of them
briefly as following:
1) Finite Difference Method: The finite difference
method is a numerical procedure which solves a partial
differential equation by discretizing the continuous phy-
sical domain into a discrete finite difference grid, appro-
ximating the individual exact partial derivatives in the
partial differential equations by algebraic finite diffe-
rence approximations (i.e. FDA), substituting the FDA’s
into the partial differential equations to obtain an alge-

M. KUMAR ET AL.1331
braic finite difference equation(FDE), and solving the
resulting algebraic finite difference equations for the
dependent variable.
For detailed study on finite difference method, we en-
courage to readers to consult the references [10-19].
2) Finite Volume Method: The finite volume me-
thod is a method for representing and evaluating partial
differential equations in the form of algebraic equations.
Similar to the finite difference method, values are calcu-
lated at discrete places on a meshed geometry. “Finite
volume” refers to the small volume surrounding each
node point on a mesh. In the finite volume method, volu-
me integrals in a partial differential equation that contain
a divergence term are converted to surface integrals, us-
ing the divergence theorem. These terms are then eva-
luated as fluxes at the surfaces of each finite volume.
Because the flux entering a given volume is identical to
that leaving the adjacent volume, these methods are con-
servative. The method is used in many computational
fluid dynamics packages
One advantage of the finite volume method over finite
difference methods is that it does not require a structured
mesh (although a structured mesh can also be used). Fur-
thermore, the finite volume method is preferable to other
methods as a result of the fact that boundary conditions
can be applied non-invasively. This is true because the
values of the conserved variables are located within the
volume element, and not at nodes or surfaces. Finite vo-
lume methods are especially powerful on coarse nonu-
niform grids and in calculations where the mesh moves
to track interfaces or shocks.
For more details on finite volume method, we refer to
[20-22].
3) Finite Element Method: The finite element me-
thod, where functions are represented in terms of basis
functions and the partial differential equations is solved
in its integral (weak) form. In the finite element method
(FEM) the domain is partitioned in a finite set of 
elements , so that

i


=
ij

 for ,ij

and
=.
i
 Usually one takes for i triangles or
quadrangles. Then the function is approximated by

=(
hii
uax 
), where i

are functions that are poly-
nomials on each i

(i.e. piecewise polynomials). Usua-
lly the functions i

are continuous polynomials of a
low degree. Further they are constructed so that their su-
pport extends only over a small number of elements.
Now we will give detail discussion of Finite element
method.
4. Finite Element Method
The key idea: The finite element is a numerical method
like finite difference method but it is more general and
powerful in its application to real-world problems that
involve complicated physical geometry and boundary
conditions.
In FEM, a given domain is viewed as a collection of
sub-domains, and over each sub-domain the governing
equation is approximated by any of the traditional vari-
ational methods.
The main reason behind taking approximate solution
on a collection of sub-domains is the fact that it is easier
to represent a complicated function as a collection of
simple polynomials.
The method is characterized by three features:
1) The domain of the problem is represented by a co-
llection of simple sub-domains called finite elements.
The collection of finite elements is called the finite ele-
ment mesh.
2) Over each finite element, the physical process is
approximated by functions of the desired type and alge-
braic equations relating physical quantities at selective
points, called nodes of the element are developed.
3) The element equation are assembled using conti-
nuity and/or “balance” of physical quantities.
In FEM, we seek an approximation un of u in the form
=1 =1
=
nn
njj
jj
uu uc
jj



 (12)
where
j
u are the values of n at the element nodes u
j

are the interpolation function,
j
c are coefficients
that are not associated with nodes, and π
j
are the
associated approximation functions. Direct substitution
of the such approximation into the governing differential
equation does not always result, for an arbitrary choice
of the data of the problem, in a necessary and sufficient
no. of equations for the under-determined coefficients
j
u and
j
c Therefore a procedure whereby a necessary
and sufficient number of equations can be obtained is
needed.
There is only one method of finite element model of
the same problem. There can be more than one finite
element model of the same problem. The type of model
depends on the differential equations, method used to
derive the algebraic equations for the undetermined coe-
fficients over an element, and nature of the approxi-
mations function used.
4.1. Variational Principles and Methods
The idea of using a variational formulation of a boundary
value problem for its numerical solution goes back to
Lord Rayleigh (1894,1896) and Ritz (1908), see, e.g.,
Kantorovich and Krylov [13]. In Ritz’s approach the
approximation solution was sought as a finite linear com-

M. KUMAR ET AL. 1332
bination of functions such as, for instance, polynomial or
trigonometric polynomials. The use in this context of
continuous piecewise linear approximating function
based on triangulation adapted to the geometry of the
domain proposed by Courant (1943) in the paper based
on an address delivered to the American Mathematical
Society in 1941. Even though this idea had appeared
earlier, also in work by Courant himself (see Babuska
[23]), this is often thought as the starting point of the
finite element method, but the further development and
analysis of the method would occur much later.
Classical sense of the variational principle is to find
the extremum or the variables of the problem. The fun-
ctional includes all the intrinsic features of the problem
such as the governing equations, boundary conditions,
and constraints condition, if any.
In solid and structural mechanics problems, the fun-
ctional represents the total energy of the system and in
other problems it is simply an integral representation of
the governing equations.
First many problems of mechanics are posed in terms
of finding the extremum and thus by their nature, can be
formulated in terms of variational statement. Second,
there are problems that can be formulated by other means,
but these can also be formulated by means of variational
principles. Third, variational formulations form a power-
ful basis for obtaining approximate solutions to practical
problems, many of which are intractable otherwise. The
principle of minimum total potential energy, for example,
can be regarded as a substitute to the equations of equi-
librium of an elastic body as well as a basis for the de-
velopment of displacement finite element models that
can be used to determine approximate displacement and
stress fields in the body.
4.2. Variational Formulations
The classical sense of the phrase “Variational Formu-
lation” refers to the construction of a functional or vari-
ational principle that is equivalent to the governing equ-
ations of the problem. The modern use of the phrase
refers to the formulation in which the governing equa-
tions are translated into equivalent weighted integral
statements that are not necessarily equivalent to a vari-
ational principle.
The importance of variational formulation of physical
laws, in the modern or general sense of the phrase, goes
for beyond its use as simply an alternate to other formu-
lation. In fact, variational forms of the laws of continuum
physics may be only natural and rigorously correct way
to think of them. While all sufficiently smooth fields
leads to meaningful variational forms the converse is not
true. There exits physical phenomena which can be ade-
quately modeled mathematically only in a variational
setting they are nonsensical when viewed locally.
The starting point for the discussion of the finite
element method is differential equations governing the
physical phenomena under study. As such, we shall first
discuss why integral statement of differential equations
are needed.
4.2.1. Need for Weighted-Integral Statements
The weighted-integral statement are required in order to
generate the necessary and sufficient number of algebraic
equations to solve for the parameters
j
c in Equation
(13) of approximate solution.
=1
() =.
N
N
jj
j
ux uc

 (13)
The use of integral statement is equivalent to the
governing differential equation is necessitated by the fact
that substitution of Equation (13) into the governing
differential does not always results in the required
number of linearly independent algebraic equations for
the unknown coefficients .
j
c One way to insure that
there are exactly the same number of equations as
there are unknowns is to require weighted integrals of the
error in the equation to be zero. We can require the
approximate solution to satisfy the given differential
equation in the weighted integral sense,
n
u
d=0,wRx

 (14)
where is called residual.
R
4.2.2. Linear and Bilinear Functional
Consider the integral expression
d
()=(,,)d, = (), =.
d
b
a
u
IuFxuuxuuxu
x


(15)
For a given real function is a real
number. Therefore,
=(), ()uuxIu
I
can be viewed as an operator that
transforms functions into real numbers, and such
operators called “functionals”.
()ux
Example 4.1
2
d
()=()()().
d
b
a
u
I
uPxqxuPu
x




a (16)
dd
(,)=(,)(,)dd d.
dd
uv
I
uvPxyqxyvxyQu s
xx






(17)
A functional is said to be linear in u if and only
if it satisfies the relation
()lu
()=()(lu vlu lv).




 (18)
for any real numbers

and

and dependent vari-
able and u.v

M. KUMAR ET AL.1333
am2 Exple 4.
1) ()= ()
bd ()().
a
I
uf x
ux qxub
2)

(,)=(,)(,)dd.
I
uvf xyuqxyvxy


unction is said to be bilinear if it is linear
in
,)
A bilinear form is said to be symmetric in its
ar
(19)
. 2. 3. V ari a tio nal Op erator and First Variation
A f(,)Buv
rgumen each of its ats u and :v

12 1
,= (,)Bu uvBu   
2
(,)v Buv 
linear in first argument.

121 2
,=(,)(Bu vvBuvBuv   
linear in second argument.
(,)Buv
f guments u and v i
( , )Buv =(,), for all , .Bvu uv
4
Consider the function (,, ).
F
xuu
 For any fixed value
of the independent variable ,
x
F
depends on u and
.
u The change v

in u wheer

is a constantuan-
and v is a fution of
q
tity nc
x
is called the variation of
u and isfined by u de

u=.v

 (20)

is called variational operator.
The variation u

of a function u represents an admi-
ssible change in t function ()ux at a fixed value of
the independent variable .
he
x
is specified value
cannot be varied. Thus, the riatio of a function u is
zero there because the specified on the boundary, the
variation of u is zero there because the specified value
cannot be vaed. Thus, the variation of a function u
should satisfy the homogeneous form of the boundary
conditions for u
If u
nva
ri
=(, ,)
F
Fxu vuv



 .
 (21)
Fundamental lemma of variational c
fu
(22)
holds for any arbitrary function
alculus: The
ndamental lemma of variations can be stated as follows
for any integrable function (),Gx if the statement
b()()d =0,

aGxx x
(), (,),
x
xab


.
tal lemma is as fo
then
llow-
in
it follows that ()=0Gx in (,)ab
General stateundament of fmen
g:
If ()
x

is arbitrary in and <<axb ()a

is arbi-
trary the
(23)
because
n the statement
()d
b()()=0
=0 in << and ()=0
aGxxBa a
GaxbBa
 


()
x

is independent of ().a

turaldar nditions
po-
(24)
These conditions are also known as Dirichle
m
on, which require specifi-
ca
4.2.4. Na and Essential Bouny Co
Essential boundary conditions which require v and
ssible its derivatives to vanish at the boundary. Thus, we
have
ˆ
Specify = 0, or = on the boundary.vuu
t or geo-
etric boundary conditions.
Natural boundary conditi
tion of the coefficient of v (and possibly its deriva-
tives). Thus we have
Specify =, on the boundary.
FQ
u


 (25)
Natural boundary conditions are also known as Neu-
m
nction such that
(26)
ann boundary conditions.
Example 4.3 Finding a fu=()uux
()=, ()=
ab
au ubu and u
()=(,(),())d is a minimum
b
a
IuFxux u xx


d
(,,)==0 in <<.
d
FF
F
xuuax b
uxu





 (27)
The necessary condition for
I
to attain a minimum
yields
dd()
d
b
b
ab
aa
()=0.
F F
vxvQvaQvb
uxu u
 
 

 


 
 


(28)
Now suppose that
F
F
u



and v are selected such that
=0 for =,=0 for =.
ab
FF
Qvxa Qv xb
uu

 
 
 


 
(29)
Then using the fundamental lemma of the calcu
va
y of
th
2)
lus of
riations, we obtain the same Euler equation.
Equations in (29) are satisfied identically for an
e following combination:
1) ()=0, ()=0.va vb
()=0, =0.
b
ab
F
va Q
u



3) =0, ()=0.
aa
FQvb
u




4) =0, =0.
aa bb
FF
QQ
uu

 
 
nsider the problem of finding defined on a
tw
Co (,),uv
o dimensional region

such that thollowing fun-
ctional is to be minimized
(,)=(, ,,
e f
:
, ,,,)dd
xxyy
I
uv Fxyu
vu v uvxy
 (30)
with condition

M. KUMAR ET AL. 1334
ˆ
= so that =0 on uu u

 (31)
ˆ
= so that =0 on vv v

 (32)
= 0on
xy
xy
FF
uu





(33)
=0on .
xy
xy
FF
vv



 (34)
Equations (31) and (32) represent the essential bo-
undary condition and Equations (33) and (34) represent
the natural boundary conditions. The pair of elements
(u,v) are called the primary variables and
= and =
x
xy yx
xy
y
F
QQ
xyv v
FFF


 
 


 
(35)
are secondary variables.
.2.5. Weak Form
d to be a weighted integral statement
.2.6. Ritz Method
e coefficients
4
Weak form is define
of a differential equation in which the differentiation is
transformed the dependent variable to the weight func-
tion such that all natural boundary conditions of the pro-
blem are also introduced.
4
In Ritz method, th
j
c
w
of the approxi-
mations are determined using the eak form of the
problem, and hence choice of weight functions is re-
stricted to the approximation function =.
j
w

Consider the variational problem resulting from the
w
(36)
for all sufficiently differentiable fun
v i
eak form: find the solution u such that
(,)=()Bwu lw
ctions w that satis-
fy the homogeneous form of the specified essential boun-
dary conditions on .u In general, (,)B
 can be un-
symmetric in w and and l is li the problem
in (36) is equi alent to minimzation of the quadratic
functional
unear,
1
()=(,) ().
2
I
uBuulu (37)
In Ritz method, we seek an approxima
(3
(38)
where the constants
tion solution to
6) in the form of a finite series
()=(
N
Ux c 
0
=1
) (),
Nj j
j
x x  
,
j
c called the Ritz coefficients are
determined such that (36) holds for each
= (=1:).
i
wiN

The function
j

and 0,

at
called approximation fun-
ctions are chosenuch th s
N
U satisfies the specified
essential boundary conditions.
N


0
=1
,=(), (=1
ij j i
j
BcliN 


 :).
Since is linear in , we have (, )B

N
u
=1
(, )=()(, )
ijjiij
j
BclB

 


=1
=,(=1:)
N
ij ji
j
K
cFi N
 (39)
=(,),=()(
iji ji
KBFlB,)
i j
 
. (40)
The algebraic equations in Equation (39) can be ex-
pressed in matrix form as:
[]{}=[] or =
K
cFKcF. (41)
4.2.7. Appro ximation Functions
Let approximate solution is sought in the fo
ed ary con-
ditions is then
rm
=1 j
and suppose that the specifi essential bound
()=
Nj j
Ux c 
()
Nx
00
()=,uxu
N
U
undary
mu satisfy the con-
dition a bo point
st
00N0
()=Ux u at=:
x
x
00
=1
( .
N
jj
j
cxu 
Since
)=

j
c
t easy
are unk be determined,
it is no to choose
nown parameters to
()
j
x

we cans
suchhat the above rela-
tion holf then select all
t
ds. I0,u
j

such that
0
()
j
x

and satisfy the condition 0 ()=0.
N
Ux
0000
=1
()=() ()
N
Njj
j
xcxx  
 (42) U
000 0
=1 =1
=() ()
NN
jj jj
jj
ucxu cx  

=0 (43)
,
j
c by choosing which is satisfied, for arbitrary
0
()=0.
jx

If all specified etial boundary ssen
0
condi-
tions are homogeneous, then

is taken to be zero and
j

must still satisfy the same conditions, 0 ()=0,
jx

=1: .jN Note that the requirement that w be zero at
the boundary conditions are specified is satisfied by the
ice =().
j
wx cho

4.2.8. The Method of Weighted Residuals
The meighthod of weted residual can be described in its
on generality by considering the operator equati
()= in .Au f (44)
where
A
is an operator (linear or nonlinear), often a
differential operator, acting on the d
f a k
ependent variables
and isnown function of the independent variables.
In the weighted residual method, the solution u is
approximated, in much the same way as in Ritz method
by the expression
00
()=() ()
N
Njj
Uxcxx  
 (45)
=1 j

M. KUMAR ET AL.1335
uirements on except that the req0

and
j

for the
weighted residual method are monge
for the Ritz method
As the name suggests, the parameter
re strint than those
0
=1
=( )=0
N
Njj
j
RAUf Acf 




. (46)
.
j
c
vani
are deter-
mined by requiring the residual to
weighted integral sense:
Rsh in the
() (,)dd.
ij
x
Rxcxy 

 (47)
where ()
i
x

are weight functionshere are following
type of well known method
. T
based on weight functions.
1) The Patrov-Galerkin method
ethod
: The weighted resi-
dual mis referred as the Petrov-Galerkin method
when .
ii

 When operator
A
is linear Equation
(47) is simplified to the form


0
=1
()d=()d
N
ijj
ji
A
xcfAx  


 (48)

=1
= or (=)
N
ij ji
j
A
cF AcF

where .
(49)
=()d
iji j
A
Ax 


2) The Galerkin method: If the weight function i

is chosen to be equal to the approximation function i

- the weethighted residual mod is better known as Ga
leinrkin method. The algebraic equations of the Galerk
approximation are
=
A
cF (50)
where


0, i
=(
iji )d, =()d j i
A
A 

xF fA


 x
 
the app
required
roximation functions are used in Galerkin method
to be higher order than those in the Ritz method.
For further readings we suggest to see [24-28].
3) The least-square method: In the least-square me-
thod, we determine the parameters
j
c by minimizing
the integral of the square of the residual.
2(,)d=0
j
i
Rxcx
c


d=0.
i
RRx
c


 (51)
Here weight function =.
ii
R
c



4) The collocation method: In the collocation me-
ate solution thod, we seek an approxim
N
U to Equation
(44) in the form of
N
U
Nby quirinsidual to vanish identically at selected points
reg the re
=(,,)
iiii
x
yz x
(=1:)iN in the domain

(,)=0 (=1:)
i
Rci Nx. (52)
The selection of points i x is crucial a
j
in obtaining
well-condi tions and ultimately in
obtaining an accurate son. The col
can be shown to be a spfied case o
w
tioned system ofqua e
lutio
eci
location method
f Equation (47)
ith


=,
i
i
 xx wher()e

x is the Dirac Delta
fun- ctions which is defined by


()d=()ff
 

xx x (53)
with he weighctions, the weighted
residual statement becomes
this choice of tt fun


(,)d =0
ij
Rc 

xxxx
(,)=0.
ij
Rcx
The steps involved in finite element method of a pro-
blem:
1) Discretization of the given domain into a collection
of prescribed finite element.
ts.
properties needed for pro-
bl
ion of element equations for all typical ele-
m
ial equation over the typical element.
(a) Construct the finite element mesh of the prescribed
elemen
(b) Number of nodes and elements.
(c) Generate the geometry
em.
2) Derivat
ent in the mesh:
(a) Construct the variational formulation of the given
different
(b) Assume that a typical dependent variable u is of
the form
=1
=
N
ii
i
uu


and substitute it into step 2(a) to obtain element equation
in the form
.[][]=[]
eee
K
uF
(c) Select, if already available in the literature, or
derive element interpolation functions i

and compute
the element matrices.
3) Assembly of element equations to obtain the equa-
tio
e results.
ite element method.
5.
In this section, we derive error estimates for the finite
n of whole problem.
4) Imposition of the boundary conditions of the pro-
blem.
5) Solution of the assembled equation.
6) Postprocessing of th
Authors can view the references [29-47] for more stu-
dy on fin
Error Estimate

M. KUMAR ET AL. 1336
ersion of the Generalized
, gives the uniqueness of the
quation and gives a first estimate
f the error. This theorem is in fact only applicable when
ings and p
e element solution. This usually
m
nerating the stiffness ma-
tri
timates in local (elemental) norms may also pro-
vi
depend
th
error estimates: A new flux is cal-
cu
n error estimates: Interpolation error
bo
Th
ols for their numerical solution available in the
damental ideas and techniques in fi-
ite difference and finite element methods have resem-
d by Department of
cience and Technology, New Delhi, Government of
e to thank the anonymous
viewers for their valuable comments and suggestions to
aux Derivees Partielles de
athematique,” American Journal of Mathe-
, No. 3, 1890, pp. 211-294.
element method. The discrete v
Lax-Milgram theorem
solution to the discrete e
o
we use finite dimensional subspaces of our original Hil-
bert spaces. We have a more general case when we don’t
have such subspaces or when the operators in the vari-
ational equation are replaced by approximations (for
instance by quadrature). We give also error estimates for
this case. The error estimates depend on how good we
can interpolate elements of Banach spaces subspaces of
these Banach spaces, so we have to discuss the inter-
polation theory in Banach spaces, preceded by a nece-
ssary discussion of the formalism of the finite element
method. This will give us estimates in the Sobolev norms
:;;.kkmq We will also give an estimate in the 2 L-
norm, but for this we need additional requirements on the
problem we consider.
1) The error estimate should give an accurate measure
of the discretization error for a wide range of mesh spac-
olynomial degrees.
2) The procedure should be inexpensive relative to the
cost of obtaining the finit
eans that error estimates should be calculated using
only local computations, which typically require an eff-
ort comparable to the cost of ge
x.
3) A technique that provides estimates of point-wise
errors which can subsequently be used to calculate error
measures in several norms is preferable to one that only
works in a specific norm. Point-wise error estimates and
error es
de an indications as to where solution accuracy is in-
sufficient and where refinement is needed.
A posteriori error estimates can roughly be divided
into four categories:
1) Residual error estimates: Local finite element pro-
blems are created on either an element or a sub-domain
and solved for the error estimate. The datas on
e residual of the finite element solution.
2) Flux-projection
lated by post processing the finite element solution.
This flux is smoother than the original finite element flux
and an error estimate is obtained from the difference of
the two fluxes.
3) Extrapolation error estimates: Two finite element
solutions having different orders or different meshes are
compared and their differences used to provide an error
estimate.
4) Interpolatio
unds are used with estimates of the unknown con-
stants.
6. Conclusions
e present paper gives a comprehensive overview of
the fundamentals of partial differential equations and re-
lated to
literature. Many fun
n
blance, and in some simple cases they coincide. Never-
theless, with its more systematic use of the variational
approach, its greater flexibility, and the way it more
easily lends itself to error analysis, the finite element
method has become the dominating approach for tackling
the partial differential equations together with their appli-
cations in science and engineering.
7. Acknowledgements
This research work is financially supported by the grant,
No. SR/FTP/MS-14/2007, sponsore
S
India. The authors would lik
re
improve the manuscript.
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