Least-squares solutions as solutions of a perturbation form of the Galerkin methods: Interior pointwise error estimates and pollution effect

• Authors: Jaeun Ku
• Venue: Journal of Computational and Applied Mathematics
• Year: 2013

First-order div least-squares (LS) finite element methods for second-order elliptic partial differential equations are considered. The main idea of this paper is a consideration of the LS solution u"h for the primary function u as the solution of a perturbation form of the Galerkin methods. Consequently, the results concerning the perturbation form of the Galerkin methods can be applied for the LS methods. Here, we use the pointwise error estimates for the perturbation form provided by Schatz [A.H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part II. Interior estimates, SIAM J. Numer. Anal. 38 (2000) 1269-1293] to obtain the pointwise estimates for the LS solution. The estimates consist of the local truncation error and pollution terms. We provide estimates for pollution terms. Our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation. Our estimates are valid for both smooth and nonsmooth problems. For the smooth problems, the pollution is weaker than the local truncation error. For the nonsmooth problems, we provide a criterion for determining the strength of the pollution. In particular, we identify superconvergent points for nonconvex polygonal domains.