FreeFem++ scripts for a posteriori error estimates via flux and potential reconstructions
Martin Vohralík

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Description

• scripts to be used with FreeFem++
• easy to understand and modify
• model steady partial differential equations in 2D
• guaranteed energy error upper bound (reliability)
• polynomial-degree- and parameter-robust energy error lower bound (efficiency)
• equilibrated flux reconstruction
• potential reconstruction
• mesh adaptivity
• solvers adaptivity

1. Laplace equation and conforming finite elements: a posteriori error estimates

• flux reconstruction by averaging
• equilibrated flux reconstruction by the solution of local energy minimization problems
• error and estimators distributions
• effectivity index
• robustness wrt polynomial degree
• illustrative implementation not meant for speed

description and exercices
FreeFem++ script
2. Laplace equation and conforming finite elements: mesh adaptivity

• Dörfler marking
• adaptive mesh regeneration
• convergence with respect to the number of degrees of freedom
• optimal (best-possible) error decay

description and exercices
FreeFem++ scripts ,
3. Laplace equation and nonconforming finite elements: a posteriori error estimates and mesh adaptivity

• equilibrated flux reconstruction by prescription
• potential reconstruction by averaging
• error and estimators distributions
• effectivity index
• Dörfler marking
• adaptive mesh regeneration
• convergence with respect to the number of degrees of freedom
• optimal (best-possible) error decay
• fast execution setting bool PlotMesh = 0, PlotSolAppr = 0, PlotSolApprFluxes = 0, PlotPotRec = 0, PlotApprPot = 0, PlotFluxRec = 0, PlotErr = 0, PlotEst = 0

description and exercices
FreeFem++ scripts ,
4. Monotonous and Lipschitz-continuous nonlinear equations and conforming finite elements: adaptive stopping criteria

• iterative linearization
• equilibrated flux reconstruction by the solution of local linear energy minimization problems
• discretization and linearization error components
• balancing error components (adaptive stopping criteria)

description and exercices
FreeFem++ script
5. Reaction–diffusion equation: parameter robustness

• equilibrated flux reconstruction by the solution of local energy minimization problems
• error and estimators distributions
• effectivity index
• robustness wrt model parameters
• the full script used for the numerical experiments in Smears, I., Vohralík, M. Simple and robust equilibrated flux a posteriori estimates for singularly perturbed reaction–diffusion problems. ESAIM Math. Model. Numer. Anal. 54 (2020), 1951–1973, Journal version, preprint

a simple example FreeFem++ script
the full FreeFem++ script

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