Least-squares Finite Element Methods for First-order Elliptic Systems

Least-squares principles use artificial " energy " functionals to provide a Rayleigh-Ritz-like setting for the finite element method. These function-als are defined in terms of PDE's residuals and are not unique. We show that viable methods result from reconciliation of a mathematical setting dictated by the norm-equivalence of least-squares functionals with practicality constraints dictated by the algorithmic design. We identify four universal patterns that arise in this process and develop this paradigm for first-order ADN elliptic systems. Special attention is paid to the effects that each discretization pattern has on the computational and analytic properties of finite element methods, including error estimates, conditioning of the algebraic systems and the existence of efficient preconditioners.