# Stable Mixed Finite Elements for Linear Elasticity with Thin Inclusions

## TL;DR

This paper develops and analyzes stable mixed finite element methods for linear elasticity problems involving complex geometries with thin inclusions modeled as mixed-dimensional manifolds, ensuring stability and convergence.

## Contribution

It introduces a novel mixed-dimensional formulation for elasticity with thin inclusions and proposes stable finite element discretizations with proven stability and convergence.

## Key findings

- The mixed-dimensional system is well-posed with appropriate norms.
- Finite element schemes conserve linear momentum locally.
- Error estimates demonstrate stability and convergence.

## Abstract

We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of hierarchically connected manifolds is formed which we refer to as mixed-dimensional. The governing equations with respect to linear elasticity are then defined on this mixed-dimensional geometry. The resulting system of partial differential equations is also referred to as mixed-dimensional, since functions defined on domains of multiple dimensionalities are considered in a fully coupled manner. With the use of a semi-discrete differential operator, we obtain the variational formulation of this system in terms of both displacements and stresses. The system is then analyzed and shown to be well-posed with respect to appropriately weighted norms. Numerical discretization schemes are proposed using well-known mixed finite elements in all dimensions. The schemes conserve linear momentum locally while relaxing the symmetry condition on the stress tensor. Stability and convergence are shown using a priori error estimates.