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

TL;DR

This paper develops and analyzes least-squares finite element methods with nonconforming elements for general second-order elliptic equations, including proofs of solvability, error estimators, and conditions for mesh independence.

Contribution

It introduces new least-squares formulations with nonconforming finite elements for elliptic equations, providing proofs of solvability, error estimation techniques, and conditions for mesh size independence.

Findings

Discrete solvability under small mesh size is proven.

Counterexample shows div least-squares functional lacks norm equivalence.

Proposed three-filed method is mesh-size independent and has built-in error estimation.

Abstract

In this paper, we study least-squares finite element methods (LSFEM) for general second-order elliptic equations with nonconforming finite element approximations. The equation may be indefinite. For the two-field potential-flux div LSFEM with Crouzeix-Raviart (CR) element approximation, we present three proofs of the discrete solvability under the condition that mesh size is small enough. One of the proof is based on the coerciveness of the original bilinear form. The other two are based on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. A counterexample shows that div least-squares functional does not have norm equivalence in the sum space of $H^1$ and CR finite element spaces. Thus it cannot be used as an a posteriori error estimator. Several versions of reliable and efficient error estimators are proposed for the method. We also propose a three-filed potential-flux-intensity div-curl least-squares method with general nonconforming finite element approximations. The norm equivalence in the abstract nonconforming piecewise $H^1$-space is established for the three-filed formulation on the minimal assumption of the uniqueness of the solution of the second-order elliptic equation. The three-filed div-curl nonconforming formulation thus has no restriction on the mesh size, and the least-squares functional can be used as the built-in a posteriori error estimator. Under some restrictive conditions, we also discuss a potential-flux div-curl least-squares method.