Displacement-pressure formulation and mini-element pair

Assume Countable Choice, as in the imported model and mini-element spaces. Use the model, spaces, and sign convention of Linear isotropic elasticity: displacement form and data. For finite $\lambda >0$, set $p=-\lambda \mathrm{div}u\in Q$ and write the mixed equations

$$a_{\mu }(u,v)+b(v,p)=f(v),b(u,q)-{\lambda }^{-1}(p,q{)}_{L^2(D)}=0$$

for all $v\in V$ and $q\in Q$. The pressure belongs to $Q$ by The mixed pressure has the mean-zero gauge, which uses the zero-mean-divergence result. At the endpoint $\lambda =\mathrm{\infty }$, define ${\lambda }^{-1}=0$ and impose $b(u,q)=0$; this is the incompressible constraint. At $\lambda =0$ use the primal displacement problem with $p=0$, not a mixed saddle system.

For a connected polygonal or polyhedral $D$ in dimension $d=2$ or $3$ with a conforming shape-regular affine simplicial mesh, the discrete pair $(V_h,Q_h)$ is exactly the P1-bubble/P1 mini-element pair of The conforming P1-bubble/P1 mini element. The discrete mixed equations use the same $a_{\mu },b,{\lambda }^{-1}$ and replace $V,Q$ by $V_h,Q_h$.