Abstract
In this article we consider the application of goal-oriented mesh adaptation to problems posed on complicated domains which may contain a huge number of local geometrical features, or micro-structures. Here, we exploit the composite variant of the discontinuous Galerkin finite element method based on exploiting finite element meshes consisting of arbitrarily shaped element domains. Adaptive mesh refinement is based on constructing finite element partitions of the domain consisting of agglomerated elements which belong to different levels of an underlying hierarchical tree data structure. As an example of the application of these techniques, we consider the numerical approximation of the incompressible Navier–Stokes equations. Numerical experiments highlighting the practical performance of the proposed refinement strategy will be presented.
Highlights
In recent years extensive work has been undertaken on the construction of finite element methods on general meshes consisting of polygonal and poly-Preprint submitted to Journal of Computational and Applied MathematicsFebruary 27, 2014 hedral elements
In our own work, stimulated by the articles [20, 19], we have developed the discontinuous Galerkin composite finite element method (DGCFEM) for problems posed on complicated computational domains which may contain local geometrical features, or micro-structures, see [1, 18]; for the application of these ideas to the construction of domain decomposition preconditioners, we refer to [17, 2]
The general idea of CFEMs is to construct the underlying finite element spaces based on first generating a hierarchy of meshes, such that the finest mesh does provide an accurate representation of the underlying computational domain, followed by the introduction of appropriate prolongation operators which determine how the finite element basis functions on the coarse mesh are defined in terms of those on the fine grid
Summary
Preprint submitted to Journal of Computational and Applied MathematicsFebruary 27, 2014 hedral elements. The key feature of CFEMs/DGCFEMs is that they allow for the construction of coarse finite element meshes, consisting of general element shapes, which provide an accurate description of the computational domain ⌦, even in the presence of micro-structures In this manner, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features present in ⌦. The general idea of CFEMs is to construct the underlying finite element spaces based on first generating a hierarchy of meshes, such that the finest mesh does provide an accurate representation of the underlying computational domain, followed by the introduction of appropriate prolongation operators which determine how the finite element basis functions on the coarse mesh are defined in terms of those on the fine grid In this way, CFEMs naturally lend themselves for exploitation within adaptive mesh refinement strategies in order that the discretization error is controlled in a reliable fashion.
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