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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.

Approximation in L2 norm. Duality argument.


e will use the proposition ( Toy approximation in energy norm ) to approximate the difference $u-u_{n}$ in $L^{2}$ norm.

Let $w$ be the solution of the problem MATH Let $v\in X^{n}$ . We estimate MATH Thus MATH We assume that the following condition holds.

Condition

(Energy approximation) The spaces $X^{n}$ satisfy MATH for any MATH and some small $\varepsilon$ .

Under such assumption we continue: MATH We recall MATH thus MATH We use the proposition ( Toy approximation in energy norm ): MATH and apply the condition ( Energy approximation ): MATH

Proposition

Under condition ( Energy approximation ) the solutions $u$ and $u_{n}$ of the problem ( Variational toy problem ) and ( Approximate toy problem ) satisfy the error estimate MATH





Notation. Index. Contents.


















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