A Parallel Multilevel Partition of Unity Method for Elliptic Partial Differential Equations

1. Front Matter

   Pages i-v

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2. Introduction

   • Marc Alexander Schweitzer

   Pages 1-11

3. Partition of Unity Method

   • Marc Alexander Schweitzer

   Pages 13-22

4. Treatment of Elliptic Equations

   • Marc Alexander Schweitzer

   Pages 23-49

5. Multilevel Solution of the Resulting Linear System

   • Marc Alexander Schweitzer

   Pages 51-96

6. Tree Partition of Unity Method

   • Marc Alexander Schweitzer

   Pages 97-126

7. Parallelization and Implementational Details

   • Marc Alexander Schweitzer

   Pages 127-153

8. Concluding Remarks

   • Marc Alexander Schweitzer

   Pages 155-159

9. Back Matter

   Pages 161-199

   PDF

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.

• Numerical integration
• Transformation
• construction
• differential equation
• hyperbolic equation
• meshfree method
• multilevel method
• numerical quadrature
• parallel computation
• partial differential equation
• partition of unity
• tree code
• partial differential equations

• Institut für Angewandte Mathematik, Universität Bonn, Bonn, Germany

  Marc Alexander Schweitzer

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