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

Griebel, Michael; Schweitzer, Marc Alexander

doi:10.1137/s1064827599355840

Cited by 156 publications

(92 citation statements)

References 56 publications

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“…In summary, the PPUM discretization of our model problem (12) using the space V PU on the cover C Ω is carried out in two steps: First, we estimate the regularization parameter β V PU from (17). Then, we define the weak form (14) with (15) and (16) and use Galerkin's method to set up the respective symmetric positive definite linear system Aũ =f .…”

Section: Essential Boundary Conditionsmentioning

confidence: 99%

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An adaptive hp-version of the multilevel particle–partition of unity method

Schweitzer

2009

Computer Methods in Applied Mechanics and Engineering

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Section: Essential Boundary Conditionsmentioning

confidence: 99%

“…In this paper we focus on the particle-partition of unity method (PPUM) [17,36]. The PPUM is a meshfree generalized finite element method based on the partition of unity (PU) approach developed in [2,3,15] but employs a socalled flat top PU to ensure that the constructed shape functions are linearly independent.…”

Section: Introductionmentioning

confidence: 99%

An adaptive hp-version of the multilevel particle–partition of unity method

Schweitzer

2009

Computer Methods in Applied Mechanics and Engineering

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“…Over the past two decades, the concept of partition of unity (PU) approximations has been established and developed into different types of PU-based methods for solid mechanics including the Partition of Unity method [1][2], hp clouds [3], the generalized finite element method [4], the octree partition of unity method (OctPUM) [5] and others [6][7][8][9]. PUFEMs have attracted much interest from researchers in computational solid mechanics as they offer several advantages over the conventional finite element method (FEM), such as a free choice of local approximation functions, which allows flexibility for modelling complicated problems, and the construction of high order approximations without the addition of extra nodes.…”

Section: Introductionmentioning

confidence: 99%

A new partition of unity finite element free from the linear dependence problem and possessing the delta property

Cai

Zhuang

Augarde

2010

Computer Methods in Applied Mechanics and Engineering

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Charles (2010) 'A new partition of unity nite element free from the linear dependence problem and possessing the delta property.', Computer methods in applied mechanics and engineering., 199 (17-20). pp. 1036-1043. Further information on publisher's website:http://dx.doi.org/10.1016/j.cma. 2009.11.019 Publisher's copyright statement:Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT AbstractPartition-of-unity based finite element methods (PUFEMs) have appealing capabilities for p-adaptivity and local refinement with minimal or even no remeshing of the problem domain. However, PUFEMs suffer from a number of problems that practically limit their application, namely the linear dependence (LD) problem, which leads to a singular global stiffness matrix, and the difficulty with which essential boundary conditions can be imposed due to the lack of the Kronecker delta property. In this paper we develop a new PU-based triangular element using a dual local approximation scheme by treating boundary and interior nodes separately. The present method is free from the LD problem and essential boundary conditions can be applied directly as in the FEM. The formulation uses triangular elements, however the essential idea is readily extendable to other types of meshed or meshless formulation based on a PU approximation. The computational cost of the present method is comparable to other PUFEM elements described in the literature. The proposed method can be simply understood as a PUFEM with composite shape functions possessing the delta property and appropriate compatibility.

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“…In the following, we shortly review the construction partition of unity spaces and the meshfree Galerkin discretization of an elliptic PDE, see [1,2] for details. Furthermore, we give a summary of the efficient multilevel solution of the arising linear block-system, see [3] for details.…”

Section: Partition Of Unity Methodsmentioning

confidence: 99%

“…The particle-partition of unity method (PUM) [1,2,3,4,5,8] is a meshfree Galerkin method for the numerical treatment of partial differential equations (PDE). In essence, it is a generalized finite element method (GFEM) which employs piecewise rational shape functions rather than piecewise polynomial functions.…”

Section: Introductionmentioning

confidence: 99%