A family of conforming mixed finite elements for linear elasticity on triangular grids

TL;DR

This paper introduces a new family of conforming mixed finite elements for linear elasticity on triangular grids, achieving stability and optimal error estimates without relying on Fortin operators, validated by numerical tests.

Contribution

A novel family of mixed finite elements for elasticity on triangles that ensures stability and optimal convergence without Fortin operators, using divergence characterization.

Findings

Proved well-posedness and stability of the finite element family.

Established optimal a priori error estimates.

Numerical tests confirm theoretical results.

Abstract

This paper presents a family of mixed finite elements on triangular grids for solving the classical Hellinger-Reissner mixed problem of the elasticity equations. In these elements, the matrix-valued stress field is approximated by the full $C^0$-$P_k$ space enriched by $(k-1)$ $H(\d)$ edge bubble functions on each internal edge, while the displacement field by the full discontinuous $P_{k-1}$ vector-valued space, for the polynomial degree $k\ge 3$. The main challenge is to find the correct stress finite element space matching the full $C^{-1}$-$P_{k-1}$ displacement space. The discrete stability analysis for the inf-sup condition does not rely on the usual Fortin operator, which is difficult to construct. It is done by characterizing the divergence of local stress space which covers the $P_{k-1}$ space of displacement orthogonal to the local rigid-motion. The well-posedness condition and the optimal a priori error estimate are proved for this family of finite elements. Numerical tests are presented to confirm the theoretical results.