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

Abstract

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, especially for indefinite equations. We answer both for general, possibly indefinite, second-order elliptic equations with non-homogeneous mixed boundary conditions. First, for the weights $h_F^{-1}$ on the Dirichlet and $1$ on the Neumann residual, we prove norm and error equivalences with mesh-independent constants for every discrete pair: the error in a mesh-dependent norm is equivalent to the square root of the functional plus a Dirichlet data term. The least-squares solution converges optimally in $H(\mathrm{div};Ω)\times H^{1}(Ω)$, and the functional is a reliable and efficient error estimator, with a recovered flux also for approximations from other methods, even ones violating the boundary conditions. The key idea is to apply the stability of the boundary value problem to pairs that are discrete only in their Dirichlet traces, whose $H^{1/2}$ norm an inverse inequality bounds by the weighted penalty. Second, these weights are the right ones: weaker powers, including $h_F$ on the Neumann boundary, destroy the norm equivalence, and stronger weights are admissible, since the penalty multiplies zero at a comparison function meeting the projected data. Third, for the fully discontinuous counterpart, a pure penalty method with both jumps weighted by $h_F^{-1}$, the error equivalence holds for every discrete pair, also for exact fluxes with normal traces not square integrable on interior faces.