TY - JOUR
T1 - A simple proof of coerciveness of first-order system least-squares methods for general second-order elliptic PDEs
AU - Zhang, Shun
PY - 2023/1/15
Y1 - 2023/1/15
N2 - In this paper, we present a simple proof of the coerciveness of first-order system least-squares methods for general (possibly indefinite) second-order linear elliptic PDEs under a minimal uniqueness assumption. The proof is inspired by Ku's proof [36] based on the a priori estimate of the PDE. For general linear second-order elliptic PDEs, the uniqueness, existence, and well-posedness are equivalent due to the compactness of the operator and Fredholm alternative. Thus only a minimal uniqueness assumption is assumed: the homogeneous equation has a unique zero solution. The proof presented in the paper is a straightforward and short proof using the inf-sup stability of the standard variational formulation. The proof can potentially be applied to other equations or settings once having the standard formulation's stability. As an application, we also discuss least-squares finite element methods for problems with a nonsingular H−1 right-hand side.
AB - In this paper, we present a simple proof of the coerciveness of first-order system least-squares methods for general (possibly indefinite) second-order linear elliptic PDEs under a minimal uniqueness assumption. The proof is inspired by Ku's proof [36] based on the a priori estimate of the PDE. For general linear second-order elliptic PDEs, the uniqueness, existence, and well-posedness are equivalent due to the compactness of the operator and Fredholm alternative. Thus only a minimal uniqueness assumption is assumed: the homogeneous equation has a unique zero solution. The proof presented in the paper is a straightforward and short proof using the inf-sup stability of the standard variational formulation. The proof can potentially be applied to other equations or settings once having the standard formulation's stability. As an application, we also discuss least-squares finite element methods for problems with a nonsingular H−1 right-hand side.
KW - Coerciveness
KW - General second-order elliptic PDEs
KW - Least-squares methods
KW - LSFEM
UR - http://www.scopus.com/inward/record.url?scp=85143371329&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85143371329&origin=recordpage
U2 - 10.1016/j.camwa.2022.11.021
DO - 10.1016/j.camwa.2022.11.021
M3 - 21_Publication in refereed journal
VL - 130
SP - 98
EP - 104
JO - Computers and Mathematics with Applications
JF - Computers and Mathematics with Applications
SN - 0898-1221
ER -