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

Abstract

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:

Discrete Normal Vector Field Approximation via Time Scale Calculus

“…Mimetic methods are a fundamental tool for simulations that do not change their physical properties, whose solution is suddenly variable, irregular grid structures, or long-running simulations, and it is getting more and more important [1]. Mimetic methods are generally used in logically uniform grids [2][3][4], in regular or unstructured grids [5][6][7], in triangular grids [8][9][10], and in polygonal grids [11,12].…”