scholarly journals Optimal Convergence Rates for Goal-Oriented FEM with Quadratic Goal Functional

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Roland Becker ◽  
Michael Innerberger ◽  
Dirk Praetorius
Keyword(s):  
Dual Problem ◽  
Convergence Rates ◽  
Linear Convergence ◽  
Elliptic Pde ◽  
Optimal Convergence ◽  
Optimal Convergence Rates ◽  
Adaptive Fem ◽  
Numer Anal

AbstractWe consider a linear elliptic PDE and a quadratic goal functional. The goal-oriented adaptive FEM algorithm (GOAFEM) solves the primal as well as a dual problem, where the goal functional is always linearized around the discrete primal solution at hand. We show that the marking strategy proposed in [M. Feischl, D. Praetorius and K. G. van der Zee, An abstract analysis of optimal goal-oriented adaptivity, SIAM J. Numer. Anal.54 (2016), 3, 1423–1448] for a linear goal functional is also optimal for quadratic goal functionals, i.e., GOAFEM leads to linear convergence with optimal convergence rates.

Mixed Schemes for Fourth-Order DIV Equations

2019 ◽  
Vol 19 (2) ◽  
pp. 341-357
Author(s):  
Ronghong Fan ◽  
Yanru Liu ◽  
Shuo Zhang
Keyword(s):  
Theoretical Analysis ◽  
Fourth Order ◽  
Convergence Rates ◽  
Mixed Formulation ◽  
Optimal Convergence ◽  
Mixed Formulations ◽  
Optimal Convergence Rates

AbstractIn this paper, stable mixed formulations are designed and analyzed for the quad div problems under two frameworks presented in [23] and [22], respectively. Analogue discretizations are given with respect to the mixed formulation, and optimal convergence rates are observed, which confirm the theoretical analysis.

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