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.
 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.
 a. H-tilde spaces.
 b. Convergence of finite elements with nonsmooth initial condition.
 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.

## H-tilde spaces.

otivated by the proposition ( Eigenvalues of symmetric elliptic operator ) we make the following definition.

Definition

(Eigenfunctions of Laplacian) For a bounded set with boundary we introduce a nondecreasing sequence such that for some and the functions form a basis in .

Definition

(H-tilde spaces) For we introduce the Hilbert spaces according to the relationships The following proposition shows that is independent of the choice of the basis .

The proposition ( Parseval equality ) illuminates the convergence of the series .

Proposition

(Characterization of H-tilde spaces) For we have where the boundary condition is understood in the sense of the proposition ( Trace theorem ). The norms and are equivalent.

Proof

Let's introduce the convenience notation

First, we show that for a we have . This would prove because is dense in . Indeed, where we calculate for every term of the sum by the definition ( Eigenfunctions of Laplacian ) by the proposition ( Green formula )-2 and on Hence, we continue by the proposition ( Parseval equality ) and by the proposition ( Green formula )-1 Thus, . We also extract for reference below.

We now prove using the same steps as above: by with The prove of is similar.

We prove the inclusion as follows. For (forming dense set in ) we have by the above calculations then by the proposition ( Boundary elliptic regularity ) The boundary condition at is evident.

The case is similar.

 Notation. Index. Contents.