Marc Alexander Schweitzer

A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic, and Hyperbolic PDEs

In this paper, we present a meshless discretization technique for instationary convection-diffusion problems. It is based on operator splitting, the method of characteristics, and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used as an h-version or a p-version. Even for general particle distributions, the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid. We discuss the implementational aspects of the proposed method. Furthermore, we present the results of numerical examples, where we considered instationary convection-diffusion, instationary diffusion, linear advection, and elliptic problems.

A Particle-Partition of Unity Method--Part II: Efficient Cover Construction and Reliable Integration

In this paper we present a meshfree discretization technique based only on a set of irregularly spaced points

A Particle-Partition of Unity Method Part V: Boundary Conditions

x_i \in \mathbb R^d

A parallel multilevel partition of unity method for elliptic partial differential equations

and the partition of unity approach. In this sequel to [M. Griebel and M. A. Schweitzer, SIAM J. Sci. Comput., 22 (2000), pp. 853--890] we focus on the cover construction and its interplay with the integration problem arising in a Galerkin discretization. We present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme. Here, we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency. The use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly. Now the overall computational costs are dominated by the number of the integration cells. We present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency. In fact, the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method, yet the presented method is completely independent of a mesh. Moreover, our method is applicable to general domains and allows for the construction of approximations of any order and regularity.

Partition of Unity Method

In this sequel to [12, 13, 14, 15] we focus on the implementation of Dirichlet boundary conditions in our partition of unity method. The treatment of essential boundary conditions with meshfree Galerkin methods is not an easy task due to the non-interpolatory character of the shape functions. Here, the use of an almost forgotten method due to Nitsche from the 1970’s allows us to overcome these problems at virtually no extra computational costs. The method is applicable to general point distributions and leads to positive definite linear systems. The results of our numerical experiments, where we consider discretizations with several million degrees of freedom in two and three dimensions, clearly show that we achieve the optimal convergence rates for regular and singular solutions with the (adaptive) h-version and (augmented) p-version.

A Particle-Partition of Unity Method--Part III: A Multilevel Solver

1 Introduction.- 2 Partition of Unity Method.- 2.1 Construction of a Partition of Unity Space.- 2.2 Properties.- 2.3 Basic Convergence Theory.- 3 Treatment of Elliptic Equations.- 3.1 Galerkin Discretization.- 3.2 Boundary Conditions.- 3.3 Numerical Results.- 4 Multilevel Solution of the Resulting Linear System.- 4.1 Multilevel Iterative Solvers.- 4.2 Multilevel Partition of Unity Method.- 4.3 Numerical Results.- 5 Tree Partition of Unity Method.- 5.1 Single Level Cover Construction.- 5.2 Construction of a Sequence of PUM Spaces.- 5.3 Numerical Results.- 6 Parallelization and Implementational Details.- 6.1 Parallel Data Structures.- 6.2 Parallel Tree Partition of Unity Method.- 6.3 Numerical Results.- 7 Concluding Remarks.- Treatment of other Types of Equations.- A.1 Parabolic Equations.- A.2 Hyperbolic Equations.- Transformation of Keys.- Color Plates.- References.

An Algebraic Multigrid Method for Linear Elasticity

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= {ω 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.

Flow Field Clustering via Algebraic Multigrid

In this sequel to part I [SIAM J. Sci. Comput., 22 (2000), pp. 853--890] and part II [SIAM J. Sci. Comput., 23 (2002), pp. 1655--1682] we focus on the efficient solution of the linear block-systems arising from a Galerkin discretization of an elliptic partial differential equation of second order with the partition of unity method (PUM). We present a cheap multilevel solver for partition of unity (PU) discretizations of any order. The shape functions of a PUM are products of piecewise rational PU functions

A Particle-Partition of Unity Method-Part IV: Parallelization

\varphi_i

Coarse grid classification: a parallel coarsening scheme for algebraic multigrid methods

with