A Parallel Multilevel Partition Of Unity Method For Elliptic Partial Differential Equations (lecture Notes In Computational Science And Engineering)
by Marc Alexander Schweitzer /
2003 / English / PDF
8.8 MB Download
the solution or its gradient. These new discretization techniques
are promising approaches to overcome the severe problem of
mesh-generation. Furthermore, the easy coupling of meshfree
discretizations of continuous phenomena to dis crete particle
models and the straightforward Lagrangian treatment of PDEs via
these techniques make them very interesting from a practical as
well as a theoretical point of view. Generally speaking, there are
two different types of meshfree approaches; first, the classical
particle methods [104, 105, 107, 108] and second, meshfree
discretizations based on data fitting techniques [13, 39].
Traditional parti cle methods stem from physics applications like
Boltzmann equations [3, 50] and are also of great interest in the
mathematical modeling community since many applications nowadays
require the use of molecular and atomistic mod els (for instance
in semi-conductor design). Note however that these methods are
Lagrangian methods; i. e. , they are based On a time-dependent
formulation or conservation law and can be applied only within this
context. In a particle method we use a discrete set of points to
discretize the domain of interest and the solution at a certain
time. The PDE is then transformed into equa tions of motion for
the discrete particles such that the particles can be moved via
these equations. After time discretization of the equations of
motion we obtain a certain particle distribution for every time
step.
the solution or its gradient. These new discretization techniques
are promising approaches to overcome the severe problem of
mesh-generation. Furthermore, the easy coupling of meshfree
discretizations of continuous phenomena to dis crete particle
models and the straightforward Lagrangian treatment of PDEs via
these techniques make them very interesting from a practical as
well as a theoretical point of view. Generally speaking, there are
two different types of meshfree approaches; first, the classical
particle methods [104, 105, 107, 108] and second, meshfree
discretizations based on data fitting techniques [13, 39].
Traditional parti cle methods stem from physics applications like
Boltzmann equations [3, 50] and are also of great interest in the
mathematical modeling community since many applications nowadays
require the use of molecular and atomistic mod els (for instance
in semi-conductor design). Note however that these methods are
Lagrangian methods; i. e. , they are based On a time-dependent
formulation or conservation law and can be applied only within this
context. In a particle method we use a discrete set of points to
discretize the domain of interest and the solution at a certain
time. The PDE is then transformed into equa tions of motion for
the discrete particles such that the particles can be moved via
these equations. After time discretization of the equations of
motion we obtain a certain particle distribution for every time
step.
