First-order least-squares formulation of the Dirichlet Poisson problem

Assume Countable Choice. Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be bounded connected Lipschitz and $f\in L^2(\mathrm{\Omega };\mathbb{R})$. Solve $-\mathrm{\Delta }u=f$, $u\in H_0^1(\mathrm{\Omega })$; set $\sigma =-\mathrm{\nabla }u$, so $\mathrm{div}\sigma =f$. For a finite conforming shape-regular triangular mesh, define $X_h=R{T}_0({\mathcal{T}}_h)\times (P_1({\mathcal{T}}_h)\cap H_0^1(\mathrm{\Omega }))$ and $J_h({\sigma }_h,u_h)=\mathrm{\parallel }{\sigma }_h+\mathrm{\nabla }{u}_h{\mathrm{\parallel }}_2^2+\mathrm{\parallel }\mathrm{div}{\sigma }_h-f{\mathrm{\parallel }}_2^2$. Here $R{T}_0(K)=P_0(K{)}^2+x{P}_0(K)$ and $P_1$ is continuous piecewise affine. The real Sobolev-space and zero-boundary conventions are Integer-order Sobolev spaces and their norms and Zero-boundary Sobolev space as a norm closure; H(div), mesh, and RT conventions are The real H(div) space and Affine simplicial meshes and RT/BDM spaces. This is the plan-spec PE30 sign convention; negating the design flux converts its residuals to these and preserves the squared objective. Integrals are exact; geometry, quadrature and roundoff errors are excluded.