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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.
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.

Finite elements for Poisson equation with Dirichlet boundary conditions.


roblem

(Poisson equation with Dirichlet boundary condition) Find the function MATH , MATH such that MATH for some function MATH . The $U$ is assumed to be a bounded domain with a $C^{\infty}$ boundary.

We multiply the equation $-\Delta u=f$ with a smooth function MATH , integrate over the domain $U$ and apply the proposition ( Green formula ). We arrive to the following weak formulation (see the section ( Elliptic PDE section ) for review of general theory):

Problem

(Poisson equation weak formulation 1) Find the function MATH such that MATH

Proposition

(Existence of weak solution of the Poisson equation) There exists a solution of the problem ( Poisson equation weak formulation 1 ) for MATH .

Proof

Use the proposition ( Existence of weak solution for elliptic Dirichlet problem 2 ) combined with the fact MATH and the proposition ( Weak maximal principle 1 ).

Proposition

(Elliptic regularity for Poisson equation) For a weak solution $u$ of the problem ( Poisson equation weak formulation 1 ) and $m=0,1,...$ we have MATH whenever the norm MATH is finite.

Proof

Combine the propositions ( Boundary elliptic regularity ) and ( Elliptic boundedness of inverse ).

Condition

(Finite dimensional approximation 1) We assume existence of a sequence of finite dimensional subspaces MATH of increasing dimensionality with the following property MATH where the real valued parameter $h$ is small and the parameter $s$ is an integer MATH for some $r\geq2$ . The $C$ is a constant.

We introduce the notation MATH for some basis of $S_{h}$ for every $h$ .

Condition

(Finite dimensional approximation 2) We assume existence of a family MATH of interpolation operators MATH with the property MATH

Problem

(Galerkin approximation 1) We define the finite dimensional approximation to the solution of the problem ( Poisson equation weak formulation 1 ) as the solution $u_{h}$ of the problem MATH

Proposition

There exists a unique solution $u_{h}$ of the problem ( Galerkin approximation 1 ).

Proof

The problem ( Galerkin approximation 1 ) may be rewritten as MATH where the $u$ is the solution of the problem ( Poisson equation weak formulation 1 ). Therefore the $u_{h}$ is the projection of $u$ on the subspace $S_{h}$ with respect to the inner product MATH in MATH . Consequently, the $u_{h}$ exists and is unique.

Definition

(Elliptic Ritz projection) We define the projection MATH according to the rule MATH

Definition

(Orthogonal L2 projection) We denote MATH the orthogonal projection on $S_{h}$ with respect to the MATH scalar product.

Proposition

Ritz projection has the following property: MATH

Proof

We set $\chi=R_{h}v$ in the definition ( Elliptic Ritz projection ): MATH Hence MATH

Proposition

(Galerkin convergence 1) Assume that the condition ( Finite dimensional approximation 1 ) holds and $u$ , $u_{h}$ are solutions of the problems ( Poisson equation weak formulation 1 ),( Galerkin approximation 1 ) respectively. We have MATH for some constant $C$ and MATH .

Proof

We pointed out in the proof of the proposition ( Galerkin approximation 1 ) that the $u_{h}$ is the projection of $u$ on $S_{h}$ with respect to the inner product MATH in MATH . Hence, MATH and according to the condition ( Finite dimensional approximation 1 ) MATH This proves the second inequality of this proposition.

We estimate MATH Let MATH be such that MATH . We continue MATH The MATH is orthogonal to $S_{h}$ . We denote $\psi_{h}$ the best approximation of $\psi$ in the sense of the condition ( Finite dimensional approximation 1 ) and continue MATH We apply the second inequality of this proposition: MATH We apply the condition ( Finite dimensional approximation 1 ) with $s=2$ : MATH We invoke the proposition ( Elliptic regularity for Poisson equation ) for MATH with $m=0$ : MATH : MATH We arrived at MATH This is the first inequality.

Proposition

(Ritz projection convergence 1) Assume that the condition ( Finite dimensional approximation 1 ) holds. We have for MATH : MATH for some constant $C$ and MATH .

Proof

The statement is a consequence of the propositions ( Galerkin convergence 1 ) and ( Existence of weak solution of the Poisson equation ) because $R_{h}u=u_{h}$ .





Notation. Index. Contents.


















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