1
UNIVERSITY EXAMINATIONS
January / February 2025
APM4810
An Introduction to the Finite Element Method
Examiners:
First: PROF E.F. DOUNGMO GOUFO
Second: DR Z. Ali
100 Marks
3 Hours
Partially open book and online examination, which you have to write within 3
hours and submit online through the link: https://myexams.unisa.ac.za/portal
Exam notes are permissible.
Use of a non-programmable pocket calculator is allowed
This web based examination remains the property of the University of South
Africa and may not be distributed from the Unisa platform.
This examination allows attachment documents only as part of your submission.
Save frequently while working.
Declaration: I have neither given nor received aid on this examination.
Answer All Questions and submit within the stipulated timeframe.
Late or email submission will not be accepted.
This paper consists of 4 pages.
ALL CALCULATIONS MUST BE SHOWN.
[TURN OVER]
, 2 APM4810
January/February 2025
QUESTION 1
In order to apply the Finite Element Method to real life problems, we have to study problems in two and three
dimensions. Give a comprehensive summary of the Finite Element Method in two dimensions.
[20]
QUESTION 2
Consider scalar linear elliptic equation of second order
Pn ∂ ∂u
) = f, in Ω ⊂ Rn
− i,j=1 ∂x (aij ∂x
(1) i j
u = 0 on ∂Ω,
where the coefficients aij = aij (x) are bounded functions and there exist γ > 0 such that
n
X
(2) γ|ξ|2 ≤ aij ξi ξj , ∀x ∈ Ω, ∀ξ ∈ Rn .
i,j=1
Convert the problem (1) into a general problem given by a continuous bilinear form a(·, ·) and a continuous linear
form L in a Hilbert space V. Show that the bilinear form a is also coercive.
[20]
QUESTION 3
Given the variational problem
[a1 (∂1 v)2 + a2 (∂2 v)2 + a3 (∂1 v − ∂2 v)2 − 2f v]dx → min!
with a1 , a2 , a3 > 0, find the associated Euler differential equation and give the difference star (also called stencil).
Hint: Choose
sh = v ∈ C Ω ; v is linear in every triangle and v = 0 on ∂Ω
so that in every triangle v ∈ sh has the form v (x, y) = a + bx + cy, and is uniquely defined by its values at the three
vertices of the triangle.
[15]
QUESTION 4
Prove the following Lemma: (Céa’s Lemma)
Suppose the bilinear form a is V –elliptic with H0m (Ω) ⊂ V ⊂ H m (Ω). In addition, suppose u and uh are solutions
of the variational problem in V and Sh ⊂ V , respectively. Then ku − uh km ≤ αc inf vh ∈Sh ku − vh km , where k·km
is the standard Sobolev norm on H m (Ω) .
[15]
[TURN OVER]
, 3 APM4810
January/February 2025
QUESTION 5
(5.1) Consider the boundary value problem (6)
00
−u + u = f on (0, `)
0
u (0) = u (`) = 0.
The interval (0, `) is divided into n elements of length h by the nodes
x0 = 0, x1 = h, ..., xn = nh = `.
Assume that piecewise linear basis functions are used and that the corresponding finite element problem
is given by
Kū = F.
Define the components of K, ū and F in terms of u, f and the basis functions.
(5.2) Consider the boundary value problem (9)
−u00 + u = f on (0, `)
u (0) = 4
u0 (`) = 3.
Assume that piecewise linear basis functions are used and that the corresponding finite element problem
is given by
Lū = G.
Find the components of L and G in terms of K and F.
[15]
QUESTION 6
Find the variational formulation of the following problems:
(6.1) (7)
00
−u + u = f on (0, `)
u (0) = 0
0
−u (`) − u (`) = 4.
ALSO: Show that the variational problem has at most one solution for problem (6.1).
(6.2) (4)
u(4) − u00 + u = f on (0, `)
u (0) = u00 (0) = 0
u0 (`) = u000 (`) = 0.
(6.3) (4)
∂t u (x, t) = ∂x2 u (x, t) + f (x, t) for x ∈ (0, `) , t>0
u (0, t) = ∂x u (`, t) = 0 for t>0
u (x, 0) = g (x) for x ∈ (0, `) .
[15]
, 4 APM4810
January/February 2025
TOTAL MARKS: [100]
c
UNISA 2024
UNIVERSITY EXAMINATIONS
January / February 2025
APM4810
An Introduction to the Finite Element Method
Examiners:
First: PROF E.F. DOUNGMO GOUFO
Second: DR Z. Ali
100 Marks
3 Hours
Partially open book and online examination, which you have to write within 3
hours and submit online through the link: https://myexams.unisa.ac.za/portal
Exam notes are permissible.
Use of a non-programmable pocket calculator is allowed
This web based examination remains the property of the University of South
Africa and may not be distributed from the Unisa platform.
This examination allows attachment documents only as part of your submission.
Save frequently while working.
Declaration: I have neither given nor received aid on this examination.
Answer All Questions and submit within the stipulated timeframe.
Late or email submission will not be accepted.
This paper consists of 4 pages.
ALL CALCULATIONS MUST BE SHOWN.
[TURN OVER]
, 2 APM4810
January/February 2025
QUESTION 1
In order to apply the Finite Element Method to real life problems, we have to study problems in two and three
dimensions. Give a comprehensive summary of the Finite Element Method in two dimensions.
[20]
QUESTION 2
Consider scalar linear elliptic equation of second order
Pn ∂ ∂u
) = f, in Ω ⊂ Rn
− i,j=1 ∂x (aij ∂x
(1) i j
u = 0 on ∂Ω,
where the coefficients aij = aij (x) are bounded functions and there exist γ > 0 such that
n
X
(2) γ|ξ|2 ≤ aij ξi ξj , ∀x ∈ Ω, ∀ξ ∈ Rn .
i,j=1
Convert the problem (1) into a general problem given by a continuous bilinear form a(·, ·) and a continuous linear
form L in a Hilbert space V. Show that the bilinear form a is also coercive.
[20]
QUESTION 3
Given the variational problem
[a1 (∂1 v)2 + a2 (∂2 v)2 + a3 (∂1 v − ∂2 v)2 − 2f v]dx → min!
with a1 , a2 , a3 > 0, find the associated Euler differential equation and give the difference star (also called stencil).
Hint: Choose
sh = v ∈ C Ω ; v is linear in every triangle and v = 0 on ∂Ω
so that in every triangle v ∈ sh has the form v (x, y) = a + bx + cy, and is uniquely defined by its values at the three
vertices of the triangle.
[15]
QUESTION 4
Prove the following Lemma: (Céa’s Lemma)
Suppose the bilinear form a is V –elliptic with H0m (Ω) ⊂ V ⊂ H m (Ω). In addition, suppose u and uh are solutions
of the variational problem in V and Sh ⊂ V , respectively. Then ku − uh km ≤ αc inf vh ∈Sh ku − vh km , where k·km
is the standard Sobolev norm on H m (Ω) .
[15]
[TURN OVER]
, 3 APM4810
January/February 2025
QUESTION 5
(5.1) Consider the boundary value problem (6)
00
−u + u = f on (0, `)
0
u (0) = u (`) = 0.
The interval (0, `) is divided into n elements of length h by the nodes
x0 = 0, x1 = h, ..., xn = nh = `.
Assume that piecewise linear basis functions are used and that the corresponding finite element problem
is given by
Kū = F.
Define the components of K, ū and F in terms of u, f and the basis functions.
(5.2) Consider the boundary value problem (9)
−u00 + u = f on (0, `)
u (0) = 4
u0 (`) = 3.
Assume that piecewise linear basis functions are used and that the corresponding finite element problem
is given by
Lū = G.
Find the components of L and G in terms of K and F.
[15]
QUESTION 6
Find the variational formulation of the following problems:
(6.1) (7)
00
−u + u = f on (0, `)
u (0) = 0
0
−u (`) − u (`) = 4.
ALSO: Show that the variational problem has at most one solution for problem (6.1).
(6.2) (4)
u(4) − u00 + u = f on (0, `)
u (0) = u00 (0) = 0
u0 (`) = u000 (`) = 0.
(6.3) (4)
∂t u (x, t) = ∂x2 u (x, t) + f (x, t) for x ∈ (0, `) , t>0
u (0, t) = ∂x u (`, t) = 0 for t>0
u (x, 0) = g (x) for x ∈ (0, `) .
[15]
, 4 APM4810
January/February 2025
TOTAL MARKS: [100]
c
UNISA 2024
