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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.
a. Elliptic problem with relaxed boundary approximation.
b. Parabolic problem with relaxed boundary approximation.
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.

Elliptic problem with relaxed boundary approximation.


e consider the problem ( Poisson equation with Dirichlet boundary condition ).

Definition

(Nitsche bilinear form) We introduce the bilinear form MATH and the norm MATH where the $\gamma$ is a positive constant and $h$ is the parameter controlling the precision of the finite dimensional approximation.

Let $u$ be a solution of the problem ( Poisson equation with Dirichlet boundary condition ). According to the proposition ( Green formula ) and the boundary conditions MATH

Condition

(Approximation in Nitsche norm) We assume that the finite dimensional spaces MATH satisfy the following conditions

1. MATH .

2. MATH

3. MATH

Problem

(Poisson equation weak formulation 2) Find the solution $u_{h}\in S_{h}$ that satisfies the condition MATH

Note that MATH

Proposition

(Nitsche form energy estimates) For a fixed $\gamma$ and sufficiently smooth $u,v$ we have MATH Assuming that the condition ( Approximation in Nitsche norm ) is satisfied, there exists numbers $\gamma>0$ and $c_{0}>0$ such that MATH

Proof

The first estimate is evident from the definitions.

We calculate the second estimate as follows: MATH We use the formula ( Cauchy inequality with epsilon ): MATH We use the condition ( Approximation in Nitsche norm )-2 as follows: MATH hence we set MATH and apply the above inequality twice: MATH We choose MATH then MATH

Proposition

(Galerkin convergence 3) Assume that MATH satisfies the condition ( Approximation in Nitsche norm ) and $u,u_{h}$ are the solutions of the problems ( Poisson equation with Dirichlet boundary condition ) and ( Poisson equation weak formulation 2 ) respectively. We have MATH for $2\leq s\leq r.$

Proof

We estimate MATH According to the proposition ( Nitsche form energy estimates ) we have MATH We use the equality MATH : MATH and use the proposition ( Nitsche form energy estimates ) again: MATH We also have MATH according to the condition ( Approximation in Nitsche norm )-3. These results conclude the proof.

Notation

(Solution operator for elliptic problem) We introduce the notation $T$ for the mapping from the function $f$ of the elliptic problem ( Poisson equation with Dirichlet boundary condition ) into its solution $u$ . The operator $T_{h}$ does the same for a particular approximation of the elliptic problem in $S_{h}$ used in the context.

Condition

(Properties of solution operator) We assume the $T$ and $T_{h}$ have the following properties:

1. $T_{h}$ is selfadjoint, positive semidefinite on MATH and positive definite on $S_{h}$ .

2. There is a positive integer $r\geq2\,$ such that MATH

Proposition

The solution operators $T$ and $T_{h}$ for the problems ( Poisson equation with Dirichlet boundary condition ) and ( Galerkin approximation 1 ) satisfy the condition ( Properties of solution operator ).

Proof

According to the definition of $T_{h}$ as the solution operator for ( Galerkin approximation 1 ) we have MATH Since $T_{h}g\in S_{h}$ for any MATH we have can apply the above equality to both positions in the scalar product MATH : MATH Hence we have the selfadjointness. We substitute $g=f$ : MATH thus $T_{h}$ is positive semidefinite on MATH . In addition, for MATH MATH Hence, $T_{h}$ is positive definite on $S_{h}$ .

The condition MATH is a consequence of the propositions ( Galerkin convergence 1 ) and ( Elliptic regularity for Poisson equation ).

Proposition

The solution operators $T$ and $T_{h}$ for the problems ( Poisson equation with Dirichlet boundary condition ) and ( Poisson equation weak formulation 2 ) satisfy the condition ( Properties of solution operator ).

Proof

The proof is similar to the previous proposition. Use the proposition ( Galerkin convergence 3 ) instead of ( Galerkin convergence 1 ).





Notation. Index. Contents.


















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