Discrete space-filling curves for adaptive mesh refinement

Abstract

For adaptive meshes that are refined and coarsened frequently partitioning according to (discrete) space-filling curves (SFCs) has been established as a fast and reliable method. Additionally to using SFCs in extreme-scale adaptive mesh refinement (AMR) codes, investigating the properties of SFCs and developing new constructions are subjects of many theoretical studies. In this talk, we provide an introduction to AMR with SFCs and develop a new self-contained mathematical framework to describe discrete SFCs. We provide definitions for the underlying structures such as refinement spaces and refinement rules. Discrete SFCs map these refinement spaces to an index set and satisfy certain locality properties. Our construction is independent of a particular geometric embedding. We are certain that this definition provides the basis for new theoretical and practical insights. As a first application, we present a cross-product construction of SFCs that allows us to construct new SFCs from existing ones, e.g. a new curve for prism shaped elements.