Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
1. Calculational Linear Algebra.
2. Wavelet Analysis.
3. Finite element method.
A. Tutorial introduction into finite element method.
a. Variational formulation, essential and natural boundary conditions.
b. Ritz-Galerkin approximation.
c. Convergence of approximate solution. Energy norm argument.
d. Approximation in L2 norm. Duality argument.
e. Example of finite dimensional subspace construction.
f. Adaptive approximation.
B. Finite elements for Poisson equation with Dirichlet boundary conditions.
C. Finite elements for Heat equation with Dirichlet boundary conditions.
D. Finite elements for Heat equation with Neumann boundary conditions.
E. Relaxed boundary conditions for approximation spaces.
F. Convergence of finite elements applied to nonsmooth data.
G. Convergence of finite elements for generic parabolic operator.
4. Construction of approximation spaces.
5. Time discretization.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Variational formulation, essential and natural boundary conditions.

e present equivalent formulation of the problem ( Toy problem ).

Let MATH be a smooth function. We multiply the equation MATH with $v\left( x\right) $ and integrate over $\left( 0,1\right) $ : MATH We perform integration by parts MATH Hence, we require that $v$ would satisfy the condition MATH . Following tradition of the section ( Elliptic PDE section ) we use the notation MATH We arrive to the following formulation.


(Variational toy problem) For a given MATH find a function MATH such that MATH


(Equivalence of toy problems) If $u$ solves the problem ( Variational toy problem ) and MATH then $u$ also satisfies ( Toy problem ).


We start from ( Variational toy problem ) and perform the integration by parts in opposite direction: MATH thus MATH Therefore the term MATH must vanish. By contradiction, if it does not vanish the we choose a sequence of $v\left( x\right) $ to blow up in a shrinking neighborhood of $1$ and remain constant elsewhere and thus obtain a contradiction. Hence, MATH But then MATH by a similar argument.


The boundary condition MATH of the problem ( Toy problem ) is explicitly incorporated in ( Variational toy problem ) as the requirement MATH . Such condition is called "essential" boundary condition. The condition MATH is incorporated implicitly. It is called "natural" boundary condition.

Notation. Index. Contents.

Copyright 2007