Mimetic Finite Difference Methods for Diffusion Equations on Unstructured Triangular Grid

Ganzha, Victor; Liska, R.; Shashkov, Mikhail; Zenger, Christoph

doi:10.1007/978-3-642-18775-9_34

Cited by 2 publications

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“…We refer to [9,10] for the derivation of matrix M E (see also [6] for the derivation of M E in the case of a general polygon). Some minimal approximation properties for this inner product are required to obtain the MFD method with the optimal convergence rate.…”

Section: Mimetic Finite Difference Methodsmentioning

confidence: 99%

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A node reconnection algorithm for mimetic finite difference discretizations of elliptic equations on triangular meshes

Váchal

Berndt²,

Lipnikov³

et al. 2005

Communications in Mathematical Sciences

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Abstract. Most efficient adaptive mesh methods employ only a few strategies, including local mesh refinement (h-adaptation), movement of mesh nodes (r-adaptation), and node reconnection (c-adaptation). Despite of its simplicity, node reconnection is the least popular of the three. However, using only node reconnection the discretization error can be significantly reduced. In this paper, we develop and numerically analyze a new c-adaptation algorithm for mimetic finite difference discretizations of elliptic equations on triangular meshes. Our algorithm is based on a new error indicator for such discretizations, which can also be used for unstructured general polygonal meshes. We demonstrate the efficiency of our new algorithm with numerical examples.Key words. adaptive mesh method, node reconnection, mimetic finite difference, tensor coefficient MSC2000 subject classifications. 65N50, 65N06 IntroductionThe predictions and insights gained from numerical simulations can only be as good as the underlying physical models and their discrete approximations. One of the approaches that can be used to increase the accuracy of discrete approximations is to employ adaptive meshes. The strategies that are used for the generation of adaptive meshes can be divided into three categories: local mesh refinement (h-adaptation; for early work in the context of finite element discretizations, see [1,18]), movement of mesh nodes (r-adaptation; see, for example, [4,15,20]), and node reconnection (c-adaptation; e.g. [19]). Note that r-adaptation does not change the connectivity of the mesh, while both h-adaptation and c-adaptation do, by the introduction of new mesh nodes and edges, and by changing the connectivity of the mesh, respectively. Despite their simplicity, node reconnection mesh adaptation methods are the least popular of the three.A robust and flexible mesh adaptation method must leverage more than one adaptation strategy (see, e.g., [15,7]). However, it is useful to analyze each strategy separately. For example, there exists a large body of literature that is concerned exclusively with the analysis of the local mesh refinement strategy. In this paper, we consider a pure node reconnection strategy.In our node reconnection adaptation strategy, we fix the positions of mesh nodes and modify only the topology. For triangular meshes, the topological changes are reduced to swapping of mesh edges. In particular, a given edge has two adjacent triangles that form a quadrilateral for which this edge is a diagonal. Deleting this edge and introducing a new edge that coincides with the other diagonal of the patch is referred to as edge swapping. One of the goals of this paper is to show that even with such a simple approach, the discretization error can be significantly decreased.We consider the model elliptic boundary value problem:

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Section: Mimetic Finite Difference Methodsmentioning

confidence: 99%

“…The MFD method has been successfully applied to many different systems of first-order PDEs on simplicial [13,12] and polygonal meshes [14,6]. In particular, for descriptions of MFD methods for the diffusion equation on unstructured triangular grids see [9,10].…”

Section: Introductionmentioning

confidence: 99%