### Project Title

Agglomeration and refinement of polytopic meshes for virtual element and discontinuous Galerkin finite element methods

### Abstract

Recently there has been significant work developing numerical methods on meshes of polytopic elements, such as discontinuous Galerkin finite element methods (DGFEM) and virtual element methods (VEM). Polytopal meshes can be defined directly, or constructed by agglomerating existing meshes of standard elements. Several methods for agglomeration already exists, which either attempt to maintain element geometry, or are based on optimal graph splitting techniques of the dual graph for the mesh. The later technique is advantageous for splitting meshes for high-performance computing. However, neither technique takes into account the results from the numerical analysis of the methods being utilised. For example, the analysis of DGFEM on polytopal elements utilises inverse estimates requiring either a limit on the number of edges, or a condition on simplexes contained within the element. In this project, we will develop an agglomeration technique based on metrics related to the analysis of DGFEM and VEM, accounting for certain a priori knowledge; e.g. domain geometry details. Furthermore, we will also develop adaptive mesh refinement using these agglomeration techniques.

### Aims

The aim of this project is to develop new algorithms for creating agglomerated meshes for use in discontinuous Galerkin finite element (DGFEM) or virtual element (VEM) methods. Agglomerated meshes consist of general polygonal or polyhedral element, where the element is constructed by joining (agglomerating) existing elements such as simplices (triangles or tetrahedra). Existing agglomeration techniques either aim to optimise the shape of the agglomerated elements, e.g., [Jones & Vassilevski, 2001] and [Kraus & Synka, 2004], or are based on optimal graph splitting techniques, e.g., [Karypis & Kumar, 1999]. Neither of these methods are designed to optimise the resulting DGFEM or VEM and, as such, may construct sub-optimal meshes with larger than ideal error. Therefore, the aim of this project is to design algorithms for constructing agglomerated meshes that

- improve the DGFEM and VEM methods defined on the mesh by considering the analysis of the numerical method,
- optimise the mesh based on a priori knowledge (such as domain geometry), and
- are extendable to space-time meshes for time-dependent problems.

Research into adaptive refinement of DGFEM and VEM based on a posteriori error estimates exists, which attempt to modify meshes in order to reduce the error. Therefore, this project will also develop algorithms for modifying the resulting agglomerated meshes based on this a posteriori knowledge, which can

- combine or split agglomerated elements based on local error estimates, and
- combine or split agglomerated elements to track moving details in time-dependent problems.

Completing our aims successfully will provide meshes that improve the numerical solution of DGFEM and VEM on agglomerated meshes. The ability to construct good agglomerated meshes is useful for methods or problems that require multiple nested meshes of different refinement levels, such as solving multi-scale problems and multigrid methods, and physical problems requiring complex meshes. These methods have many uses in the solving of physical problems, such as in aerodynamics/fluid dynamics, biological modelling, medical imaging, geophysics, weather forecasting, and climate change models.