Crouzeix--Raviart--Marini realisation of computable a priori $L^2$-error bounds on anisotropic meshes

We consider conforming $\text{Pone}$ and lowest-order Crouzeix–Raviart (CR) approximations of the Dirichlet Poisson problem in two and three dimensions. We obtain computable a priori $L^2$-error bounds on anisotropic simplicial meshes without a global $H^2$-regularity assumption. For piecewise constant data, the Marini relation gives an equilibrated lowest-order Raviart–Thomas (RT) flux from the scalar CR solution. The square of the resulting Marini constant is the largest generalised eigenvalue of a problem involving only scalar conforming and CR matrices. Combined with the elementwise Poincaré constant, it controls both the conforming and CR errors. The CR $L^2$ estimate uses positivity of the discrete CR–conforming gap operator rather than a nonconforming Aubin–Nitsche argument. We also derive an exact decomposition of the Marini defect into the CR–conforming energy gap and an explicit geometric term with no aspect-ratio factor. On the algebraically graded L-shaped mesh family considered here, $q>3\mathrm{/}2$ yields ${\kappa }_{M,h}=O(h)$ and computable $O(h^2)$ $L^2$-operator bounds.