# Hybrid coupling of CG and HDG discretizations based on Nitsche's method

## TL;DR

This paper introduces a minimally-intrusive hybrid coupling strategy for CG and HDG discretizations using Nitsche's method, enabling effective simulation of thermal and elastic problems with multiple materials.

## Contribution

It presents a novel coupling approach based solely on the HDG hybrid variable, preserving the core structures of CG and HDG methods and facilitating integration into existing libraries.

## Key findings

- Achieves optimal convergence of stress fields.
- Demonstrates superconvergence of displacement.
- Handles multiple materials with different compressibility.

## Abstract

A strategy to couple continuous Galerkin (CG) and hybridizable discontinuous Galerkin (HDG) discretizations based only on the HDG hybrid variable is presented for linear thermal and elastic problems. The hybrid CG-HDG coupling exploits the definition of the numerical flux and the trace of the solution on the mesh faces to impose the transmission conditions between the CG and HDG subdomains. The continuity of the solution is imposed in the CG problem via Nitsche's method, whereas the equilibrium of the flux at the interface is naturally enforced as a Neumann condition in the HDG global problem. The proposed strategy does not affect the core structure of CG and HDG discretizations. In fact, the resulting formulation leads to a minimally-intrusive coupling, suitable to be integrated in existing CG and HDG libraries. Numerical experiments in two and three dimensions show optimal global convergence of the stress and superconvergence of the displacement field, locking-free approximation, as well as the potential to treat structural problems of engineering interest featuring multiple materials with compressible and nearly incompressible behaviors.