A coarsening algorithm on adaptive red-green-blue refined meshes
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A coarsening algorithm on adaptive red-green-blue refined meshes Stefan A. Funken1 · Anja Schmidt1 Received: 20 January 2020 / Accepted: 17 August 2020 / © The Author(s) 2020
Abstract Adaptive meshing is a fundamental component of adaptive finite element methods. This includes refining and coarsening meshes locally. In this work, we are concerned with the red-green-blue refinement strategy in two dimensions and its counterpartcoarsening. In general, coarsening algorithms are mostly based on an explicitly given refinement history. In this work, we present a coarsening algorithm on adaptive redgreen-blue meshes in two dimensions without explicitly knowing the refinement history. To this end, we examine the local structure of these meshes, find an easyto-verify criterion to adaptively coarsen red-green-blue meshes, and prove that this criterion generates meshes with the desired properties. We present a MATLAB implementation built on the red-green-blue refinement routine of the ameshref-package (Funken and Schmidt 2018, 2019). Keywords Coarsening · Meshes · Grids · Adaptivity · Refinement · Adaptive finite element method · RGB · Red-green-blue Mathematics Subject Classification (2010) 65M50
1 Introduction Adaptive meshing is a popular tool to efficiently solve partial differential equations where solutions exhibit local singularities [18]. In time-dependent problems, singularities, interfaces, and forces may move or change in time. This requires coarsening meshes locally. Otherwise, the algorithm’s efficiency would decrease with time since Anja Schmidt
[email protected] Stefan A. Funken [email protected] 1
Institute for Numerical Mathematics, Helmholtzstraße 20, 89081, Ulm, Germany
Numerical Algorithms
degrees of freedom needed for an earlier time step are not released as the singularity or interface progresses. To this end, it is common to deploy coarsening algorithms to maintain the adaptive efficiency [2, 23]. Furthermore, coarsening routines are used in multigrid techniques where a sequence of coarse and fine meshes is needed [17, 19]. Local geometric refinement is a major part of adaptive meshing. The goal is to reduce the element size by adding further nodes to a given mesh. Several refinement strategies are known which have desired properties and are therefore well suited for adaptive meshing. An overview and a list of public code are provided by Schneiders in [24]. Local coarsening is the counterpart of local refinement and is thus also an important part of adaptive meshing. There are different approaches to coarsening. Local coarsening refers to deleting nodes from a given mesh to increase the element size. Possible approaches are based on edge collapsing [1, 19], centroidal Voronoi tessellations [26], or the refinement history [2, 7, 16, 23]. The latter approach aims to invert the refinement based on the refinement history. Desired properties such as the inscribed ball condition [8] are automatically fulfilled during coarsening. The first two approaches, in contrast, do not use the refinem
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