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.
 a. Variational formulation, essential and natural boundary conditions.
 b. Ritz-Galerkin approximation.
 c. Convergence of approximate solution. Energy norm argument.
 d. Approximation in L2 norm. Duality argument.
 e. Example of finite dimensional subspace construction.
 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.

## Variational formulation, essential and natural boundary conditions.

e present equivalent formulation of the problem ( Toy problem ).

Let be a smooth function. We multiply the equation with and integrate over : We perform integration by parts Hence, we require that would satisfy the condition . Following tradition of the section ( Elliptic PDE section ) we use the notation We arrive to the following formulation.

Problem

(Variational toy problem) For a given find a function such that

Proposition

(Equivalence of toy problems) If solves the problem ( Variational toy problem ) and then also satisfies ( Toy problem ).

Proof

We start from ( Variational toy problem ) and perform the integration by parts in opposite direction: thus Therefore the term must vanish. By contradiction, if it does not vanish the we choose a sequence of to blow up in a shrinking neighborhood of and remain constant elsewhere and thus obtain a contradiction. Hence, But then by a similar argument.

Remark

The boundary condition of the problem ( Toy problem ) is explicitly incorporated in ( Variational toy problem ) as the requirement . Such condition is called "essential" boundary condition. The condition is incorporated implicitly. It is called "natural" boundary condition.

 Notation. Index. Contents.