Least-Squares Finite Element Methods with Weakly Imposed Boundary Conditions: Norm and Error Equivalences, Over-Penalization, and Fully Discontinuous Discretizations

Weakly imposed boundary conditions in first-order system least-squares methods go back to Jespersen (1977), but two questions have remained open for decades: the right weights, especially for the Neumann residual, whose natural norm $H^{-1/2}$ suggests $h_F$, and norm and error equivalences, especia