Several Proofs of Coerciveness of First-Order System Least-Squares Methods for General Second-Order Elliptic PDEs

Abstract

In this paper, we present three versions of proofs of the coercivity for first-order system least-squares methods for second-order elliptic PDEs. The first version is based on the a priori error estimate of the PDEs, which has the weakest assumption. For the second and third proofs, a sufficient condition on the coefficients ensuring the coercivity of the standard variational formulation is assumed. The second proof is a simple direct proof and the third proof is based on a lemma introduced in the discontinuous Petrov-Galerkin method. By pointing out the advantages and limitations of different proofs, we hope that the paper will provide a guide for future proofs. As an application, we also discuss least-squares finite element methods for problems with H −1 righthand side.

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

“…The uniqueness, existence, and well-posedness are equivalent for the linear second-order elliptic equation, for example, see discussions in various standard PDE books [3,39,37]. A detailed discussion of the following theorem with two proofs can also be found in Theorem 2.3 of [59]. Some more discussion can also be found in the introductions of [26,27].…”

“…The following norm equivalence is standard. Different proofs can be found in [18,9,11,45,59]. Theorem 3.1.…”

“…With the help of the inf-sup condition of a h (•, •) (6.7), we can prove the coerciveness following the argument in [59].…”

“…The proof also corrects a small error in the original [17] when handling mixed boundary conditions. In the third proof, we present a very short proof by combining the new proof presented in [59] and the discrete stability result proved in [26] for CR approximation of the general elliptic equation. Different from the conforming LSFEM in [18], in the second and third proofs, the div CR-LSFEM requires the same regularity and sufficient small mesh size assumptions as the non-least-squares (conforming, nonconforming, and mixed) methods [55,56,26,27].…”

“…The results require some regularity assumption and that the mesh size of the discretization is small enough. On the other hand, once the coerciveness of the least-squares method is proved, see [18,6,9,11,45,59], the LSFEM with H 1 -H(div) conforming finite element approximations is automatic coercive, without the restriction on the mesh size.…”