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Quantitative Analysis
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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.
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.

Spacial discretization for Heat equation with Dirichlet boundary conditions.


roblem

(Galerkin approximation 2) We define the finite dimensional approximation to the solution of the problem ( Heat equation weak formulation 1 ) as the solution $u_{h}$ of the problem MATH where the $g_{h}$ is some $S_{h}$ -based approximation of $g$ .

Proposition

There exists a solution of the problem ( Galerkin approximation 2 ).

Proof

Follow the procedure of the section ( Galerkin approximation for parabolic Dirichlet problem ). Under a reasonable choice of the basis MATH for the space MATH the problem ( Galerkin approximation 2 ) becomes a Cauchy problem for a system of ODEs that always have a solution.

Proposition

(Galerkin convergence 2) Let $u_{h}$ and $u$ be the solutions of the problems ( Galerkin approximation 2 ) and ( Heat equation with Dirichlet boundary condition ) respectively. Assume that $g|_{\partial U}=0$ . We have for $t\geq0$ :

1. MATH

2. MATH

Proof

(1) We introduce the notations MATH so that MATH We estimate, according to the proposition ( Ritz projection convergence 1 ), MATH We put the MATH over the relationship MATH and conclude MATH To estimate the $\theta$ we write MATH and substitute the definition of $\theta$ : MATH We want to remove everything involving $\nabla$ . Hence we substitute the relationships MATH and MATH MATH We substitute the relationship MATH : MATH or MATH We have MATH , hence, we substitute $\chi=\theta$ in the last relationship and obtain MATH Consequently MATH and we continue deriving the consequences: MATH MATH MATH According to the initial conditions for $u$ and $u_{h}$ , MATH and we use the proposition ( Ritz projection convergence 1 ) in the second term: MATH The statement now follows from the obtained results MATH MATH

Proof

(2) We have MATH According to the proposition ( Ritz projection convergence 1 ) MATH We return to the relationship MATH from the first part of this proof and substitute $\chi=\theta_{t}$ : MATH We substitute MATH and MATH (proposition ( Cauchy inequality )): MATH and we continue deriving consequences: MATH MATH MATH and the second estimate follows after application of the proposition ( Ritz projection convergence 1 ) to the term MATH .





Notation. Index. Contents.


















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