Rate-optimal goal-oriented adaptive FEM for semilinear elliptic PDEs. (15th July 2022)

• Record Type: Journal Article
• Authors: Becker, Roland; Brunner, Maximilian; Innerberger, Michael; Melenk, Jens Markus; Praetorius, Dirk
• Is Part Of: Computers & mathematics with applications. Volume 118(2022)
• Page Start: 18
• Page End: 35
• Publication Date: 2022-07-15
• Subjects: Adaptive finite element method -- Semilinear PDEs -- Quantity of interest -- A posteriori error estimation -- Goal-oriented adaptive algorithm -- Optimal convergence rates; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2022.05.008
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: We formulate and analyze a goal-oriented adaptive finite element method for a semilinear elliptic PDE and a linear goal functional. The discretization is based on finite elements of arbitrary (but fixed) polynomial degree and involves a linearized dual problem. The linearization is part of the proposed algorithm, which employs a marking strategy different to that of standard adaptive finite element methods. Moreover, unlike the well-known dual-weighted residual strategy, the analyzed error estimators are completely computable. We prove linear convergence and, for the first time in the context of goal-oriented adaptivity for nonlinear PDEs, optimal algebraic convergence rates. In particular, the analysis does not require a sufficiently fine initial mesh.