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.

## Weak formulation for Heat equation with Dirichlet boundary conditions.

e multiply the equation with a smooth function , integrate over the domain and apply the proposition ( Green formula ). We arrive to the following weak formulation (see the section ( Parabolic PDE section ) for review of general theory):

Problem

(Heat equation weak formulation 1) Find the function such that

Problem

(Existence of weak solution of the Heat equation) There exists a solution of the problem ( Heat equation weak formulation 1 ).

Proof

The statement is a direct consequence of the proposition ( Existence of weak solution for the parabolic Dirichlet problem ).

Proposition

(Parabolic regularity for heat equation) If the functions and satisfy the compatibility conditions of the proposition ( Parabolic regularity 2 ) then the solution of the problem ( Heat equation weak formulation 1 ) satisfies the estimates for some constant .

 Notation. Index. Contents.