Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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.
 a. Weak formulation for Neumann boundary conditions. Natural and essential 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.

## Weak formulation for Neumann boundary conditions. Natural and essential boundary conditions.

e 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 ( Parabolic PDE section ) for review of general theory):

Problem

(Heat equation weak formulation 2) Find the function such that

Note that we no longer require that the test functions would satisfy the boundary condition. The boundary condition is now a part of the weak formulation. Indeed, if we apply the proposition ( Green formula ) to the problem ( Heat equation weak formulation 2 ) then we get The Neumann boundary condition is therefore called a "natural boundary condition" and the Dirichlet condition is called an "essential boundary condition".

 Notation. Index. Contents.