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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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
The key objective of Scientific Computing is to provide and further develop a new promising research tool for natural and engineering scientists, Numerical Simulation. Here, we use knowledge from theoretical as well as experimental research to set up virtual experiments on a computer. This approach yields a third novel way to describe the complex questions arising from our natural environment and can help in their investigation. The three main areas of research we can distinguish in scientific computing are: mathematical modeling, the development of appropriate numerical methods, and the efficient implementation of the simulation environment on a (parallel) computer.
Marc Alexander Schweitzer

Chapter 2. Partition of Unity Method

Abstract
In the following, we present a general partition of unity method (PUM) for a meshfree discretization of an elliptic partial differential equation. The approach is roughly as follows: The discretization is stated in terms of points x i only. To obtain a trial and test space VPU, a patch or volume ω i ⊂ ℝ d is attached to each point x i such that the union of these patches form an open cover CΩ= i } of the domain Ω, i.e. \( \bar{\Omega } \subset \cup {\omega _{i}} \). Now, with the help of weight functions W i : ℝ d → ℝ with supp \(({W_i}) = \overline {{\omega _i}}\) local shape functions φ i are constructed by Shepard’s method. The functions φ i form a partition of unity (PU). Then, each partition of unity function φ i is multiplied with a sequence of local approximation functions ψ i n to assemble higher order shape functions. These product functions φ i ψ i n are finally plugged into the weak form to set up a linear system of equations via a Galerkin discretization, which we discuss in the next chapter.
Marc Alexander Schweitzer

Chapter 3. Treatment of Elliptic Equations

Abstract
We are interested in the approximate solution of an elliptic boundary value problem of the type
$$Lu = f{\text{ in }}\Omega\subset {\mathbb{R}^{d}},$$
(3.1)
$$\begin{gathered} , \hfill \\ Bu = g{\text{ on }}\partial \Omega {\text{,}} \hfill \\ \end{gathered}$$
where L is a symmetric partial differential operator of second order and B expresses suitable boundary conditions. Here, we are faced with two major computational tasks: We need to discretize the partial differential operator efficiently and we need to deal with boundary conditions properly.
Marc Alexander Schweitzer

Chapter 4. Multilevel Solution of the Resulting Linear System

Abstract
In the following we focus on the solution of the large sparse linear block-system \(A\tilde u = \hat f\) where ũ denotes a coefficient vector and \(\hat f\) denotes a moment vector. This solution step is a very time consuming part of any numerical simulation. The use of an inappropriate solver can drive up the compute time as well as the storage demand dramatically.
Marc Alexander Schweitzer

Chapter 5. Tree Partition of Unity Method

Abstract
Even though all presented components of our multilevel PUM, the PU construction, the numerical quadrature scheme, and the multilevel solver, are designed for general point sets P and general covers C Ω we applied our meshfree discretization method to uniform node arrangements only; up to now. So far we did not have to deal with the challenging task of the cover construction for irregular point sets. In this chapter we now focus on this very critical issue of constructing a suitable cover C Ω for a general given point set efficiently.
Marc Alexander Schweitzer

Chapter 6. Parallelization and Implementational Details

Abstract
In this chapter we present the parallelization of our multilevel partition of unity method [57]. Our parallelization follows the data decomposition approach. Here, the main ingredients are a key-based tree implementation and a space filling curve load balancing scheme.
Marc Alexander Schweitzer

Chapter 7. Concluding Remarks

Abstract
In this monograph we have developed a truly meshfree method for the Galerkin discretization of elliptic partial differential equations. The presented partition of unity method not only allows for the approximation of a PDE in a bounded domain Ω⊂ℝ d with the classical h-version and p-version approaches but rather supports also the use of locally augmented approximation spaces \(V_i^{p_i^a} = span\left\langle {\left\{ {\psi _i^n,\Phi } \right\}} \right\rangle\).
Marc Alexander Schweitzer

Backmatter

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