2016/08/30 by Wei Gong, Ningning Yan, Gong, Wei +3
Computer Science · Engineering · Mathematics · #49J20 #65K10 #65N12 #65N15 #65N30 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1608.08699
openalex publication_date 2016/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper aims to study the convergence of adaptive finite element method for control constrained elliptic optimal control problems under L2-norm. We prove the contraction property and quasi-optimal complexity for the L2-norm errors of both the control, the state and adjoint state variables with L2-norm based AFEM, this is in contrast to and improve our previous work [13] where convergence of AFEM based on energy norm had been studied and suboptimal convergence for the control variable was obtained and observed numerically. For the discretization we use variational discretization for the control and piecewise linear and continuous finite elements for the state and adjoint state. Under mild assumptions on the initial mesh and the mesh refinement algorithm to keep the adaptive meshes sufficiently mildly graded we prove the optimal convergence of AFEM for the control problems, numerical results are provided to support our theoretical findings.