First Order System Least Squares with inhomogeneous boundary conditions

Thanks to (1.1), the bilinear form on X×X at the left hand side is bounded, symmetric, and elliptic, and the right-hand side defines a bounded functional on X . From the Lax-Milgram lemma, we conclude that (1.2), and so the least-squares problem, has a unique solution u∈X that depends continuously on f ∈Y ′. Whenever the equation Gu= f has a solution, i.e., f ∈IG (consistency), it is the unique solution of the least-squares problem. For a closed subspace Xh ⊂ X , the Galerkin solution uh ∈ Xh of (1.2), i.e., argminvh∈Xh 1 2‖Gvh− f‖ 2 Y ′ , satisfies ‖u−uh‖X . infvh∈Xh ‖u−vh‖X , only dependent on the hidden constants absorbed by theh symbol in (1.1). Because of the symmetry of the bilinear form, the Galerkin solution is conveniently computed, assuming 〈·, ·〉Y ′ can be evaluated. In view of the latter, in the setting of a boundary value problem, usually one prefers Y ′ to be (a multiple copy of) L2(Ω).