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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
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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.
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 MATH with $C^{\infty}$ boundary $\partial U$ we introduce a nondecreasing sequence MATH such that MATH for some MATH and the functions MATH form a basis in MATH .

Definition

(H-tilde spaces) For $s\geq0$ we introduce the Hilbert spaces MATH according to the relationships MATH The following proposition shows that MATH is independent of the choice of the basis MATH .

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

Proposition

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

Proof

Let's introduce the convenience notation MATH

First, we show that for a MATH we have MATH . This would prove MATH because MATH is dense in $G^{1}$ . Indeed, MATH where we calculate for every term of the sum MATH by the definition ( Eigenfunctions of Laplacian ) MATH by the proposition ( Green formula )-2 and $v=0$ on $\partial U$ MATH Hence, we continue MATH by the proposition ( Parseval equality ) MATH and by the proposition ( Green formula )-1 MATH Thus, MATH . We also extract MATH for reference below.

We now prove MATH using the same steps as above: MATH by MATH with $v:=\Delta^{p}v$ MATH The prove of MATH is similar.

We prove the inclusion MATH as follows. For MATH (forming dense set in $\tilde{H}^{2p}$ ) we have by the above calculations MATH then by the proposition ( Boundary elliptic regularity ) MATH The boundary condition $v=0$ at $\partial U$ is evident.

The case $2p+1$ is similar.





Notation. Index. Contents.


















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