School of Mathematics and Computing
Sciences,
Chalmers University of
Technology
and
Goteborg University
The dayly course progress on TMA372/MAN660, Partial Differential equations for TM, E3, GU, ..., Fall vt 2001
OBS! the room for exercise on Monday 3/12, 15-17 is:
MD6 in Mathematcal Center, and on Monday 10/12
EL43 in Electrical Engineering building.
The
Assignment 1a (ps file),
Assignment 1a (pdf file),
(handed in no later than week4).
An example note-book,
Övningsexemple i PDE1, TM (ps file), and
(pdf file),
consisting of problems from some previous exams and solutions to the
Lösningar för udda uppgifter i Övningsexemple
i PDE1 TM, (ps file), and
(pdf file),
odd problems in Övningsexemple i PDE1, are now available both
here and through the cousre homepage as well.
Here is the course lecture diary so far:
10/10; Chap. 6 in CDE: I lectured through the corresponding hand
out for chapter 6, motivating for the use of quadrature rule (chapter
5) and for solving linear system of equations.
2/11; Chap. 5: I derived some interpolation estimates and
introduced the Cardinal functions and quarature rule, cf; handouts.
6/11; Chap. 5 and 7: Proved a general quadrature rule
for polynomials. Also introduced the interative methods (Jacobi and
Gauss-Seidel) for solving linear system of equations, cf; handouts.
6/11; Chap. 8: I have introduced the finite element method
for a 1D stationary heat equation: Boundary value problem (BVP),
have proved that variational formulation (VF), boundary value problem,
and a minimization problem (MP) are equivalent. We derived also an a
priori error estimate for the finite element method in energy norm.
13/11; Chap. 8 and 9: Chapter 8:
Proof of an a posteriori error estimate
for the finite element method. Chapter 9: Introduced a initial value
problem (IVP). Fundamental solution and stability estimates for IVP.
Continuous and discontinuous Galerkin methods; in particular cG(1) and
dG(0) for IVP.
16/11; Chap. 9: Detailed cG(1) and dG(0) aposteriori and a
priori error analysis for the IVP.
20/11; Chap. 9: Continuation of the detailed cG(1) and dG(0)
aposteriori and a priori error analysis for the IVP.
23/11; Chap. 21: Poincare' inequality, Riesz representation
theorem.
27/11; Chap. 21 and 14: Chapter 21: Lax-Milgram theorem.
Chapter 14: Green's formula.
30/11; Chap. 14 and 15: Chapter 14: ended,
Chapter 15: Stability and a priori error analysis for the Poisson's equation.
4/12; Chap. 15 and 16: Chapter 15: completeing the detailed theory of a
priori estimates, a posteriori error estimates in a
one space dimensional case. Chapter 16: Stability estimates for
multidimensional heat equation.
7/12; Chap. 16: cG(1) a posteriori error analysis
for the heat equation and for a population model.
11/12; Chap. 17 and 18: Chapter 17: the multidimensional wave equation,
conservation of energy. A space-time cG(1) algorithm for the wave
equation in one space dimension (system and matrix forms).
Chapter 18: Convection-diffusion equation, a traffic flow, the
Navier-Stokes equations.
14/12: We solved some problems from previous exams.
The final exam will contain a theory question from the
following list:
Theorem 8.1; a priori error estimate for the 2 point boundary
value problems,
Theorem 8.2; a posteriori error estimate for the 2 point boundary
value problems,
Lemma 9.1; stability estimates for the dual of a general initial value
problem,
Theorem 9.2; a posteriori error estimates for cG(1), for
a general initial value
problem,
Theorem 9.3; a posteriori error estimates for dG(0), for
a general initial value
problem,
Theorem 9.4; a priori error estimates for dG(0), for
a general initial value
problem,
Theorem 21.1; the Lax-Milgram Theorem,
Theorem 14.1; piecewise linear polynomial interpolation in 2D
(give the proof only in the 1D case as we did in class for chapter 5).
Theorem 15.4; a posteriori error estimates for the Poisson's
equation,
M. Asadzadeh
Dec. 14, 2001