First-Order System \CL\CL* (FOSLL*): Scalar Elliptic Partial Differential Equations

The L2-norm version of first-order system least squares (FOSLS) attempts to reformulate a given system of partial differential equations so that applying a least-squares principle yields a functional whose bilinear part is H1-elliptic. This ellipticity means that the minimization process amounts to solving a weakly coupled system of Poisson-like scalar equations. An unfortunate limitation of the L2-norm FOSLS approach is that this product H1 equivalence generally requires sufficient smoothness of the original problem. Inverse-norm FOSLS overcomes this limitation, but at a substantial loss of real efficiency. The FOSLL* approach introduced here is a promising alternative that is based on recasting the original problem as a minimization principle involving the adjoint equations. This paper provides a theoretical foundation for the FOSLL* methodology and illustrates its performance by applying it numerically to several examples. Results for the so-called two-stage approach applied to discontinuous coefficient problems show promising robustness and optimality. Indeed, FOSLL* appears to exhibit the generality of the inverse-norm FOSLS approach while retaining the full efficiency of the L2-norm approach.