Adaptive Discontinuous Galerkin Methods on Polytopic Meshes
In this article we consider the application of discontinuous Galerkin finite element methods, defined on agglomerated meshes consisting of general polytopic elements, to the numerical approximation of partial differential equation problems posed on compli
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Abstract In this article we consider the application of discontinuous Galerkin finite element methods, defined on agglomerated meshes consisting of general polytopic elements, to the numerical approximation of partial differential equation problems posed on complicated geometries. Here, we assume that the underlying computational domain may be accurately represented by a geometry-conforming fine mesh Tfine ; the resulting coarse mesh is then constructed based on employing standard graph partitioning algorithms. To improve the accuracy of the computed numerical approximation, we consider the development of goal-oriented adaptation techniques within an automatic mesh refinement strategy. In this setting, elements marked for refinement are subdivided by locally constructing finer agglomerates; should further resolution of the underlying fine mesh Tfine be required, then adaptive refinement of Tfine will also be undertaken. As an example of the application of these techniques, we consider the numerical approximation of the linear elasticity equations for a homogeneous isotropic material. In particular, the performance of the proposed adaptive refinement algorithm is studied for the computation of the (scaled) effective Young’s modulus of a section of trabecular bone.
1 Introduction Over the last couple of decades extensive work has been undertaken on the design and mathematical analysis of numerical methods for the approximation of partial differential equations (PDEs) based on exploiting general meshes consisting of polytopic elements, i.e., polygons/polyhedra in two-/three-dimensions, respectively. In particular, we mention the Polygonal Finite Element Method [34], the Extended Finite Element Method [21], the Mimetic Finite Difference Method [10, 12, 16], the Virtual Element Method [11], the Hybrid High Order Method [19, 20], the Composite Finite Element Method [1, 24–26], and the closely related Agglomerated Discontinuous Galerkin (DG) method [5–7]. The exploitation of general polytopic
J. Collis • P. Houston () School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK e-mail: [email protected]; [email protected] © Springer International Publishing Switzerland 2016 G. Ventura, E. Benvenuti (eds.), Advances in Discretization Methods, SEMA SIMAI Springer Series 12, DOI 10.1007/978-3-319-41246-7_9
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elements offers great flexibility for mesh generation, and moreover allows for sequences of nested, successively coarser, meshes to be generated for use within multi-level solvers, such as multigrid and domain decomposition preconditioners, cf. [2, 4, 8, 22], for example. We point out that polytopic elements naturally arise when fictitious domain methods, unfitted methods or overlapping meshes are employed, cf. [13–15, 28, 30], for example. The motivation here for employing polytopic elements is very much inspired by the work undertaken by Hackbusch and Sauter on Composite Finite Element methods in the articles [25, 26]; fo
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