A short note on plain convergence of adaptive least-squares finite element methods. (15th September 2020)

• Record Type: Journal Article
• Authors: Führer, Thomas; Praetorius, Dirk
• Is Part Of: Computers & mathematics with applications. Volume 80:issue 6(2020)
• Page Start: 1619
• Page End: 1632
• Publication Date: 2020-09-15
• Subjects: Least squares finite element methods -- Adaptive algorithm -- Convergence; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2020.07.022
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: We show that adaptive least-squares finite element methods driven by the canonical least-squares functional converge under weak conditions on PDE operator, mesh-refinement, and marking strategy. Contrary to prior works, our plain convergence does neither rely on sufficiently fine initial meshes nor on severe restrictions on marking parameters. Finally, we prove that convergence is still valid if a contractive iterative solver is used to obtain the approximate solutions (e.g., the preconditioned conjugate gradient method with optimal preconditioner). The results apply within a fairly abstract framework which covers a variety of model problems.