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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.

Crank-Nicolson time discretization for the Heat equation with Dirichlet boundary conditions.


roblem

(Crank-Nicolson problem for heat equation) We introduce a time step $\tau$ , mesh MATH the time derivative approximation MATH and the averaging operation MATH . We seek the array MATH of functions MATH that satisfies the conditions MATH

Proposition

(Crank-Nicolson convergence for heat equation) Let $U^{n}$ and $u$ be the solutions of the problems ( Crank-Nicolson problem for heat equation ) 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 substitute the relationships MATH and MATH : MATH We substitute the relationship MATH taken at $t_{n-\frac{1}{2}}$ : MATH where we introduced the notation MATH We set MATH in the equality MATH and obtain MATH hence MATH or, after substitution of definitions for MATH and $\hat{\theta}^{n}$ , MATH and after cancellation for MATH : 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 The $\omega_{1}^{n}$ was estimated when proving the proposition ( Backward Euler convergence 2 ): MATH We estimate $\omega_{2}^{n}$ using the proposition ( Integral form of Taylor decomposition ) around $t_{n-\frac{1}{2}}$ (and drop the $x$ from the notation for shortness) MATH hence MATH The $\omega_{3}$ is estimated almost exactly the same way MATH The rest follows similarly to the proof of the proposition ( Backward Euler convergence 2 ).





Notation. Index. Contents.


















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