Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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 , mesh and the time derivative approximation . We seek the array of functions that satisfies the conditions

Proposition

(Backward Euler convergence 2) Let and be the solutions of the problems ( Backward Euler problem ) 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 want to remove all the spacial terms. We substitute the relationships and : We substitute the relationship taken at : where we introduced the notation We set in the equality and obtain hence We substitute the definition of : and obtain thus or 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 : We have and we apply the the proposition ( Ritz projection convergence 1 ): Thus, The estimation of is done with similar means:

 Notation. Index. Contents.