The a posteriori error estimate

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Table of contents

Introduction
General context
General properties for an a posteriori error estimate
A posteriori error analysis based on equilibrated fluxes
Motivations
Moving boundary problems
Reservoir simulation
Outline
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Bibliography
1 Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem
1.1 Introduction
1.2 Continuous and discrete settings
1.2.1 Continuous setting
1.2.2 Discrete setting
1.3 An a posteriori error estimate for the dual norm of the residual
1.3.1 Dual norm of the residual
1.3.2 General assumptions
1.3.3 A basic a posteriori error estimate
1.3.4 An a posteriori error estimate distinguishing the space, time, regularization, linearization, and quadrature errors
1.4 Balancing and stopping criteria, adaptive algorithm, and efficiency
1.4.1 Balancing and stopping criteria
1.4.2 Adaptive algorithm
1.4.3 Efficiency of the a posteriori error estimate
1.5 An a posteriori error estimate for the error in temperature and enthalpy
1.5.1 Bounding the error of the temperature and enthalpy by the dual norm of the residual
1.5.2 The a posteriori error estimate
1.6 Application to a vertex-centered finite volume discretization
1.6.1 Dual and tertial space meshes
1.6.2 The vertex-centered finite volume scheme
1.6.3 Newton linearization
1.6.4 Flux reconstruction
1.7 Numerical experiments
1.7.1 Setting
1.7.2 Computing approximately the negative norms
1.7.3 Stopping criteria
1.7.4 Balancing criteria
1.7.5 Overall performance
1.A Proofs
1.A.1 Proof of Theorem 1.4.4
1.A.2 Proof of Theorem 1.5.2
Bibliography
2 A posteriori error estimates, stopping criteria, and adaptivity for multi- phase compositional flows in porous media
2.1 Introduction
2.2 Setting
2.2.1 The multiphase compositional model
2.2.2 An implicit finite volume scheme with phase-upwind and two-point discretization of diffusive fluxes
2.3 A basic a posteriori error estimate
2.3.1 Weak solution
2.3.2 A generic approximate solution
2.3.3 Error measure
2.3.4 Flux and pressure reconstructions
2.3.5 A posteriori error estimate
2.4 Application to finite volume method and adaptivity based on distinguishing the different error components
2.4.1 Linearization and algebraic resolution
2.4.2 Approximate solution
2.4.3 Phase pressure reconstructions
2.4.4 Component flux reconstructions
2.4.5 Distinguishing the space, time, linearization, and algebraic errors
2.4.6 A fully adaptive algorithm
2.5 Numerical results
2.5.1 Common setting
2.5.2 Compressible flow in a homogeneous porous medium
2.5.3 Compressible flow in a heterogeneous porous medium
2.5.4 Five-spots pattern
Bibliography
3 A posteriori error estimates for thermal multiphase compositional flows in porous media
3.1 Introduction
3.2 Setting
3.3 Discretization of the energy equation
3.3.1 Two-point finite volume discretization
3.3.2 Linearization and algebraic resolution
3.4 Approximate solution and reconstructions
3.4.1 Postprocessing of the temperature
3.4.2 Saturations, molar fractions, and molar energy
3.4.3 H1 0 -conforming temperature reconstruction
3.4.4 H(div; )-conforming energy flux reconstructions
3.5 A posteriori error estimate
3.5.1 Weak solution
3.5.2 Error measure
3.5.3 An a posteriori error estimate distinguishing the space, time, linearization, and algebraic errors
3.5.4 Balancing and stopping criteria
3.A Application to the thermal dead oil model
3.A.1 Dead oil model
3.A.2 A posteriori error estimate for the thermal dead oil model
Bibliography
4 Steam-assisted gravity drainage: a posteriori estimates with simplified eval- uation and application of adaptive mesh refinement
4.1 Introduction
4.2 SAGD characterization and modeling
4.2.1 Common characteristics
4.2.2 Types of SAGD
4.2.3 SAGD modeling
4.2.4 Mathematical model of the thermal dead oil system
4.3 Evaluation of the estimators using a practical simplified formula
4.3.1 A general simplification formula
4.3.2 Evaluation of the estimators
4.4 SAGD test case
4.4.1 Model description
4.4.2 Initialization and production scheme
4.4.3 Model simulation
4.4.4 Approximate solution and a posteriori estimate
4.4.5 Adaptive mesh refinement
Bibliography