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 , mesh the time derivative approximation and the averaging operation . We seek the array of functions that satisfies the conditions

Proposition

(Crank-Nicolson convergence for heat equation) Let and be the solutions of the problems ( Crank-Nicolson problem for heat equation ) and ( Heat equation with Dirichlet boundary condition ) respectively. Assume that then

Proof

We split the error term as follows We estimate the components and according to the procedure of the proof of the proposition ( Galerkin convergence 2 )-1. The has exactly the same estimate

We estimate as follows: We substitute the relationships and : We substitute the relationship taken at : where we introduced the notation We set in the equality and obtain hence or, after substitution of definitions for and , and after cancellation for : We apply the last inequality repeatedly and arrive to the estimate where the is estimated as in the proof of the proposition ( Galerkin convergence 2 )-1: It remains to estimate the : The was estimated when proving the proposition ( Backward Euler convergence 2 ): We estimate using the proposition ( Integral form of Taylor decomposition ) around (and drop the from the notation for shortness) hence The is estimated almost exactly the same way The rest follows similarly to the proof of the proposition ( Backward Euler convergence 2 ).

 Notation. Index. Contents.