Due Date: April 8, 2014
1. Consider the Sturm-Liouville problem
(1 + x2 )u ) + xu
0 < x < 1,
u(1) = 2.
Transform the problem to a problem with homogeneous Dirihlet boundary condition at x = 1. Write
down the weak form for each of t
The Finite Element
Method for 2D Problems
The FE procedure to solve 2D problems is the same as that for 1D problems, as the ow
chart below demonstrates.
PDE Integration by parts weak form in V : a(u, v) = L(v)
or min F (v) Vh (nite dimensional s
Issues of the FE Method
in one space dimensions
Boundary Conditions, Implementation & the LaxMilgram Lemma
For a second order two-point BVP, typical boundary conditions (BC) include one of the
Finite Element Methods
for 1D Elliptic Problems
The nite element (FE) method was developed to solve complicated problems in
engineering, notably in elasticity and structural mechanics modeling involving elliptic PDE and complicated geometries, b
of the Finite Element
Using FE methods, we need to answer these questions:
What is the appropriate functional space V for the solution?
What is the appropriate weak or variational form of a dierential equation.
Numerical Solutions of Partial Dierential
Equations An Introduction to Finite Dierence
and Finite Element Methods
December 17, 2012
Center for Research in Scientic Computation & Department of Mathematics, North
Due February 20
1. Take n = 3, the number of variables, describe the Sobolev space H 3 (), i.e. m = 3, in terms of L2 ()
(list all terms). Also explain the inner product, the norm, the Schwartz inequality, the distance, and
Due April 22
1. Use the Lax-Milgram Lemma to show whether or not the following two-point value problem has a
u + q(x)u = f ,
0 < x < 1,
u (0) = u (1) = 0 ,
where q(x) qmin > 0. What happens if we relax the condition
Due Jan. 30
Please read the Homework Guidlines on the course webpage rst.
1. This problem deals with the following boundary value problem:
u (x) + u(x) = f (x),
u(0) = u(1) = 0.
(a) Show that the weak form (variational form) is given by