History of mathematicians
In this document we give some information of mathematicians which
work or names are used in the Finite Element part of
the course Computational Fluid Dynamics II
(a PhD course from the
JM Burgerscentrum).
1. Introduction
Many flow problems are described by the Navier-Stokes equations
Claude Louis Marie Henri Navier (1785-1836) and
George Gabriel Stokes (1819-1903).
2. Introduction to the Finite Element method
In boundary value problems a differential equation is given together
with appropriate boundary conditions, in order to make the solution
unique. There are various boundary conditions possible.
We consider a heat equation, where the required solution describes
the temperature (T). To derive the differential equation equation
the law of
Jean Baptiste Joseph Fourier (1768-1830)
is used, which the heat flux with the first derivative of the
temperature.
As boundary conditions one can prescribe the temperature (called a
Dirichlet condition
Johann Peter Gustav Lejeune Dirichlet (1805-1859))
or one can prescribe the flux, the first derivative of the
temperature (called a Neumann condition
Carl Gottfried Neumann (1832-1925)).
In this part the Gauss divergence theorem is used. Furthermore
to compute the element matrices we use the Gauss integration rule
Carl Friedrich Gauss (1777-1855).
Also Newton Cotes integration rules are used
(
Roger Cotes (1682-1716),
Isaac Newton (1642-1727)).
The weak formulation of the boundary value problem is solved numerically
by the Galerkin method (
Boris Grigorievich Galerkin (1871-1945)).
Galerkin published his finite element method in 1915.
In most applications linear or quadratic element functions are used.
The linear basisfunctions are 1 in one node and 0 in all other nodes.
This can easily described by the Kronecker delta (
Leopold Kronecker (1823-1891)).
3. Convection-diffusion equation by the Finite Element method
The discretization of the instationary convection-diffusion equation
results in a system of ordinary differential equations. A number of
methods to solve such a system are:
For convection dominated flows the Streamline Upwind Petrov-Galerkin
method is a popular method. In its derivation the Taylor
series expansion
(
Brook Taylor (1685-1731))
is used.
4. Discretization of the incompressible Navier-Stokes equations by
standard Galerking
The Navier-Stokes
equations are made dimensionless by the introduction of the Reynolds number
(
Osborne Reynolds (1842-1912)). After discretization one obtains a
non-linear system of equations. In order to solve this non-linear system
an iterative procedure is necessary. Examples of such methods are:
the Picard or the Newton-Raphson method
( Jean Picard (1620-1682),
Isaac Newton (1642-1727),
and
Joseph Raphson (1648-1715)).
5. The penalty function method
In this chapter a method is discussed which tries to solve the
Navier-Stokes equations by separating the computation of velocity and
pressure.
6. Divergence free elements
Here the velocity is decomposed in a tangential and a normal component
along the boundary instead of Cartesian components
(
René Descartes (1596-1650)).
7. The instationary Navier-Stokes equations
In the pressure correction method the pressure is solved from a
Poisson-type equation
Siméon Denis Poisson (1781-1840)).
Contact information:
Kees Vuik
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