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 , such that for some function . The is assumed to be a bounded domain with a boundary.

We multiply the equation with a smooth function , integrate over the domain 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 such that

Proposition

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

Proof

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

Proposition

(Elliptic regularity for Poisson equation) For a weak solution of the problem ( Poisson equation weak formulation 1 ) and we have whenever the norm 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 of increasing dimensionality with the following property where the real valued parameter is small and the parameter is an integer for some . The is a constant.

We introduce the notation for some basis of for every .

Condition

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

Problem

(Galerkin approximation 1) We define the finite dimensional approximation to the solution of the problem ( Poisson equation weak formulation 1 ) as the solution of the problem

Proposition

There exists a unique solution of the problem ( Galerkin approximation 1 ).

Proof

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

Definition

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

Definition

(Orthogonal L2 projection) We denote the orthogonal projection on with respect to the scalar product.

Proposition

Ritz projection has the following property:

Proof

We set in the definition ( Elliptic Ritz projection ): Hence

Proposition

(Galerkin convergence 1) Assume that the condition ( Finite dimensional approximation 1 ) holds and , are solutions of the problems ( Poisson equation weak formulation 1 ),( Galerkin approximation 1 ) respectively. We have for some constant and .

Proof

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

We estimate Let be such that . We continue The is orthogonal to . We denote the best approximation of in the sense of the condition ( Finite dimensional approximation 1 ) and continue We apply the second inequality of this proposition: We apply the condition ( Finite dimensional approximation 1 ) with : We invoke the proposition ( Elliptic regularity for Poisson equation ) for with : : We arrived at This is the first inequality.

Proposition

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

Proof

The statement is a consequence of the propositions ( Galerkin convergence 1 ) and ( Existence of weak solution of the Poisson equation ) because .

 Notation. Index. Contents.