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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
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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.
B. Finite elements for Poisson equation with Dirichlet boundary conditions.
C. Finite elements for Heat equation with Dirichlet boundary conditions.
a. Weak formulation for Heat equation with Dirichlet boundary conditions.
b. Spacial discretization for Heat equation with Dirichlet boundary conditions.
c. Backward Euler time discretization for Heat equation with Dirichlet boundary conditions.
d. Crank-Nicolson time discretization for the 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.

Backward Euler time discretization for Heat equation with Dirichlet boundary conditions.


roblem

(Backward Euler problem) We introduce a time step $\tau$ , mesh MATH and the time derivative approximation MATH . We seek the array MATH of functions MATH that satisfies the conditions MATH

Proposition

(Backward Euler convergence 2) Let $U^{n}$ and $u$ be the solutions of the problems ( Backward Euler problem ) and ( Heat equation with Dirichlet boundary condition ) respectively. Assume that MATH then MATH

Proof

We split the error term as follows MATH We estimate the components $\rho^{n}$ and $\theta^{n}$ according to the procedure of the proof of the proposition ( Galerkin convergence 2 )-1. The $\rho^{n}$ has exactly the same estimate MATH

We estimate $\theta^{n}$ as follows: MATH MATH We want to remove all the spacial $\nabla$ terms. We substitute the relationships MATH and MATH : MATH We substitute the relationship MATH taken at $t_{n}$ : MATH where we introduced the notation MATH We set $\chi=\theta^{n}$ in the equality MATH and obtain MATH hence MATH We substitute the definition of MATH : MATH and obtain MATH thus MATH or MATH We apply the last inequality repeatedly and arrive to the estimate MATH where the $\theta^{0}$ is estimated as in the proof of the proposition ( Galerkin convergence 2 )-1: MATH It remains to estimate the MATH : MATH We have MATH and we apply the the proposition ( Ritz projection convergence 1 ): MATH Thus, MATH The estimation of $\omega_{2}^{n}$ is done with similar means: MATH MATH





Notation. Index. Contents.


















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