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Table of contents
Introduction
1 An adaptive hp-renement strategy with computable guaranteed bound on the error reduction factor
1.1 Introduction
1.2 Discrete setting
1.3 The modules SOLVE, ESTIMATE, and MARK
1.3.1 The module SOLVE
1.3.2 The module ESTIMATE
1.3.3 The module MARK
1.4 The module REFINE
1.4.1 hp-decision on vertices
1.4.2 hp-decision on simplices
1.4.3 hp-renement
1.4.4 Summary of the module REFINE
1.5 Guaranteed bound on the error reduction factor
1.6 Numerical experiments
1.6.1 Smooth solution (sharp Gaussian)
1.6.2 Singular solution (L-shape domain)
1.7 Conclusions
2 An adaptive hp-renement strategy with inexact solvers and computable guaranteed bound on the error reduction factor
2.1 Introduction
2.2 Setting and notation
2.3 Guaranteed total and algebraic a posteriori error bounds
2.4 The inexact hp-adaptive algorithm
2.4.1 The module ONE_SOLVER_STEP
2.4.2 The module ESTIMATE
2.4.3 Adaptive stopping criteria for the algebraic solver
2.4.4 The module MARK
2.4.5 The module REFINE
2.5 Guaranteed bound on the error reduction
2.6 Numerical experiments
2.6.1 Smooth solution (sharp Gaussian)
2.6.2 Exponential convergence
2.6.3 Smooth solution (asymmetric wave front)
2.6.4 Singular solution (L-shape domain)
2.7 Conclusions
3 Convergence of adaptive hp-renement strategies with computable guaranteed bound on the error reduction factor
3.1 Introduction
3.2 Framework and notation
3.3 The hp-adaptive algorithm { exact setting
3.3.1 The modules SOLVE and ESTIMATE
3.3.2 The module MARK
3.3.3 The module REFINE
3.3.4 Discrete lower bound on the incremental error on marked simplices
3.4 Discrete stability of equilibrated uxes in an exact setting
3.5 The proof of convergence with an exact solver
3.6 The inexact hp-adaptive algorithm
3.6.1 Adaptive sub-loop of ONE_SOLVER_STEP and ESTIMATE
3.6.2 Adaptive stopping criterion for the algebraic solver
3.6.3 Modules MARK and REFINE
3.6.4 Discrete lower bound on the incremental error on marked simplices
3.6.5 Conditions on the adaptive stopping criterion parameter
3.7 Discrete stability of equilibrated uxes in an inexact setting
3.8 The proof of convergence with an inexact solver
3.9 Conclusions and outlook
Appendix A Implementation details of hp-AFEM
