A stabilization-free HDG for linear elasticity with strongly symmetric stress on Alfeld splits

In this paper, we design and analyze a stabilization-free hybridizable discontinuous Galerkin (HDG) method of arbitrary polynomial orders for the linear elasticity problem based on Alfeld splits, where the symmetry of stress is strongly imposed. The [Formula: see text]-conforming space is used for the approximation of the displacement and the standard polynomial space is used for the approximation of the stress. The tangential trace of displacement acts as the Lagrange multiplier. The construction of the method hinges on the observation that the non-symmetric tensor field can be symmetrized by solving local discrete Stokes problem with inf-sup stable Scott–Vogelius pairs on Alfeld splits, and this idea also facilitates the proof of the discrete [Formula: see text] stability of the displacement. The characterization of the hybridization indicates that the resulting global system only involves the trace variable on the interface and the global system is symmetric positive definite, rending the method computationally efficient. The quasi-optimal approximation (up to data-oscillation term) is established for the [Formula: see text]-error of stress, discrete [Formula: see text]-error and [Formula: see text]-error of displacement with [Formula: see text]-independent constants without requiring additional regularity assumption. To guide adaptive mesh refinement, [Formula: see text]-robust a posteriori error estimator is derived. Several numerical experiments are carried out to verify the proposed theories.