A discrete trace theory for non-conforming polytopal hybrid discretisation methods

TL;DR

This paper develops a discrete trace theory for non-conforming hybrid discretization methods on polytopal meshes, providing theoretical results and numerical verification relevant for preconditioner design.

Contribution

It introduces a discrete trace seminorm and proves trace, lifting, and truncation estimates for hybrid discretizations on polytopal meshes.

Findings

Established a discrete trace seminorm for hybrid methods

Proved trace and lifting theorems with respect to a discrete $H^1$-seminorm

Validated theoretical results through numerical experiments on spectrum analysis

Abstract

In this work we develop a discrete trace theory that spans non-conforming hybrid discretization methods and holds on polytopal meshes. A notion of a discrete trace seminorm is defined, and trace and lifting results with respect to a discrete $H^1$-seminorm on the hybrid fully discrete space are proven. Building on these results we also prove a truncation estimate for piecewise polynomials in the discrete trace seminorm. Finally, we conduct two numerical tests in which we compute the proposed discrete operators and investigate their spectrum to verify the theoretical analysis. The development of this theory is motivated by the design and analysis of preconditioners for hybrid methods, e.g., of substructuring domain decomposition type.