High order mixed finite elements with mass lumping for elasticity on triangular grids

A family of conforming mixed finite elements with mass lumping on triangular grids are presented for linear elasticity. The stress field is approximated by symmetric $H(\mathrm{d}\mathrm{i}\mathrm{v})-P_k(k\ge 3)$ polynomial tensors enriched with higher order bubbles so as to allow mass lumping, which can be viewed as the Hu-Zhang elements enriched with higher order interior bubble functions. The displacement field is approximated by $C^{-1}-P_{k-1}$ polynomial vectors enriched with higher order terms to ensure the stability condition. For both the proposed mixed elements and their mass lumping schemes, optimal error estimates are derived for the stress with $H(\mathrm{d}\mathrm{i}\mathrm{v})$ norm and the displacement with $L^2$ norm. Numerical results confirm the theoretical analysis.