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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