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In this paper, we study least-squares finite element methods (LSFEM) forgeneral second-order elliptic equations with nonconforming finite elementapproximations. The equation may be indefinite. For the two-fieldpotential-flux div LSFEM with Crouzeix-Raviart (CR) element approximation, wepresent three proofs of the discrete solvability under the condition that meshsize is small enough. One of the proof is based on the coerciveness of theoriginal bilinear form. The other two are based on the minimal assumption ofthe uniqueness of the solution of the second-order elliptic equation. Acounterexample shows that div least-squares functional does not have normequivalence in the sum space of $H^1$ and CR finite element spaces. Thus itcannot be used as an a posteriori error estimator. Several versions of reliableand efficient error estimators are proposed for the method. We also propose athree-filed potential-flux-intensity div-curl least-squares method with generalnonconforming finite element approximations. The norm equivalence in theabstract nonconforming piecewise $H^1$-space is established for the three-filedformulation on the minimal assumption of the uniqueness of the solution of thesecond-order elliptic equation. The three-filed div-curl nonconformingformulation thus has no restriction on the mesh size, and the least-squaresfunctional can be used as the built-in a posteriori error estimator. Under somerestrictive conditions, we also discuss a potential-flux div-curl least-squaresmethod.

Least-Squares Methods with Nonconforming Finite Elements for General Second-Order Elliptic Equations