8/4/2019 AFEM01

1/2

Ruhr-Universitat Bochum Winter term 2010/11Fakultat fur Mathematik Prof. Dr. R. Verfurth

Adaptive Finite Element MethodsWork sheet 1

Exercise 1. Decide which of the following functions are contained inH1(0, 1)

u(x) = 1 + x2

3 , v(x) = 4

x,

w(x) =

9x2 for 0 x < 13

,

1 for 13

x < 2

3,

3 3x for 23 x 1,

z(x) =

3x for 0 x < 13

,

9x3 for 13

x < 2

3,

6 2x for 23 x 1.

Justify your answer and eventually compute the weak derivative.

Exercise 2. Build the stiffness matrix and the load vector for thelinear finite element discretization of the Sturm-Liouville problem

pu + qu = f in (0, 1), u(0) = u(1) = 0with

p(x) = 1 + x2, q(x) = 6, f(x) = 8 + 16x

corresponding to a partition T consisting of 3 intervals of equal lengthand compute the finite element solution uT. Determine the valuesuT(0.25) and uT(0.375).

Exercise 3. Decide which of the following functions are contained inH1()

u(x, y) = 1 + x2y3 + x3y, v(x, y) = 4

x2 + y2,

w(x, y) =

9x2y for xy 0,xy4 for xy < 0,

z(x, y) =

3x2 + y2 for xy 0,9x3 + 5y for xy < 0.

Here, denotes the interior of a circle with radius 1 centred at theorigin. Justify your answer and eventually compute the first order

weak derivatives.

Exercise 4. Give the variational formulation of the differential equa-tion

2u

x2+ 2

2u

xy 3

2u

y2+ 2(x2 + y2)u = f in ,

u = 0 on .

Exercise 5. Describe the basic steps of the finite element discretizationof an elliptic differential equation.

Exercise 6. How does one handle Neumann boundary conditions in

the finite element discretization of elliptic differential equations?1

8/4/2019 AFEM01

2/2

2

Exercise 7. Build the system of linear equations corresponding to alinear finite element discretization of the differential equation

u = xy in u = 0 on

in the unit square

=

(x, y) : 0 < x, y < 1

and compute the discrete solution uT. Here, the partition T is aCourant triangulation consisting of right angled isosceles triangles withshort sides of length h = 1

3. Compute the values uT(0.25, 0.25) and

uT(0.375, 0.5).

Exercise 8. Write a program that solves the linear finite element dis-cretization of the differential equation

u = f in u = 0 on

in the unit square

=

(x, y) : 0 < x, y < 1

using a Courant triangulation consisting of right angled isosceles tri-angles with short side of length h = 1

n, n 2. Integrals should be

evaluated using a suitable quadrature formula.

Exercise 9. Compute the nodal shape functions z,1 of the triangle Kwith vertices (1, 1), (5, 2) and (3, 4)(1) using a transformation to the reference triangle,(2) using the geometrical data of K.

Exercise 10. Compute the nodal shape functions z,1 of the parallel-ogram K with vertices (1, 1), (5, 2), (7, 5) und (3, 4)(1) using a transformation to the unit square,(2) using the geometrical data of K.