The MFE method in the DD method : Jacobi iterative solver

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Table of contents

Résumé
Abstract
Acknowledgements
Introduction
1 Domain decomposition for steady diffusion in mixed formulations
1.1 The diffusion equation
1.2 Presentation of some function spaces
1.3 The two-subdomain DD case in the mixed formulation
1.3.1 Multidomain formulation with physical transmission conditions .
1.3.2 Multidomain formulation with Robin transmission conditions
1.4 The case of many subdomains using the Optimized Schwarz Method
1.4.1 Local solver of the Jacobi method
1.5 The local solver in the mixed finite element formulation
1.5.1 Continuous problem: weak mixed formulation
1.5.2 Discrete problem: approximation by the mixed finite element method
1.6 Numerical results
1.6.1 The MFEmethod in one domain with different boundary conditions
1.6.2 The MFE method in the DD method: Jacobi iterative solver .
2 Estimates and stopping criteria in steady diffusion case
2.1 Postprocessing of pk+1 h in the lowest-order Raviart–Thomas case
2.2 Concept of potential and flux reconstructions
2.2.1 Potential reconstruction
2.2.2 Subdomain potential reconstruction
2.2.3 Equilibrated flux reconstruction
2.3 General a posteriori error estimates for ˜ph ∈ H1(Th) and uh ∈ L2( )
2.4 Properties of uk+1 h and pk+1 h in at each iteration of the DD algorithm .
2.5 Potential reconstructions for the Robin DD in the MFE method
2.5.1 Potential reconstruction
2.5.2 Subdomain potential reconstruction
2.6 Flux reconstruction for the Robin DD in the MFE method
2.6.1 Construction of k+1 h ∈ H(div, )
2.6.2 Improving k+1 h to obtain the balance with the source term .
2.7 Numerical results
2.7.1 Example 1 with the Jacobi solver
2.7.2 Example 1 with the GMRES solver
2.7.3 Example 2 with the GMRES solver
3 Estimates and stopping criteria in unsteady diffusion case
3.1 The heat equation
3.2 The global-in-time Optimized Schwarz method using OSWR
3.3 Local solver of the OSWR method for the heat equation
3.4 Local solver of the heat equation in the mixed finite element formulation
3.5 Discretization using MFE in space and an implicit scheme in time
3.6 Concept of potential and flux reconstruction for the heat equation
3.6.1 Potential reconstruction
3.6.2 Subdomain potential reconstruction
3.6.3 Equilibrated flux reconstruction
3.7 General a posteriori error estimate
3.8 Potential and flux reconstructions for the global-in-time DD in the MFE method
3.8.1 Potential reconstruction
3.8.2 Subdomain potential reconstruction
3.8.3 Flux reconstruction
3.9 Numerical results
3.9.1 Model example with the Jacobi solver
3.9.2 Model example with the GMRES solver
3.9.3 Example in an industrial context using conforming time grids .
3.10 Global-in-time domain decomposition using nonconforming time grids .
3.11 A posteriori error estimates for nonconforming time grids
3.12 Numerical results
3.12.1 Example in an industrial context using nonconforming time grids
4 Estimates and stopping criteria in a two-phase flow problem
4.1 Introduction
4.2 Presentation of the problem
4.2.1 Flow between two rock types
4.2.2 Transformation of the equations and weak formulation
4.3 Space-Time Domain Decomposition Methods with Ventcell Transmission
4.4 The cell-centered finite volume scheme
4.4.1 Space-time discretization, notations, and function spaces
4.4.2 A space-time fully discrete scheme based on finite volumes in space and the backward Euler scheme in time
4.4.3 Newton linearization
4.5 Postprocessing and H1- and H(div)-conforming reconstructions
4.5.1 Discontinuous piecewise quadratic ˜ϕ k,n,m h,i and postprocessed saturation uk,n,m h,i
4.5.2 Continuous piecewise quadratic ˆ ϕk,n,m h,i and H1-conforming reconstruction sk,m hτ
4.5.3 Equilibrated flux reconstruction k,m hτ
4.6 A posteriori error estimate
4.7 An a posteriori error estimate distinguishing the space, time, linearization, and the DD errors
4.8 Stopping criteria and optimal balancing of the different error components
4.9 Numerical experiments
4.9.1 The performance of the OSWR method with adaptive stopping criteria
4.9.2 Comparison of Robin- and Ventcell-OSWR algorithm with adaptive stopping criteria
Conclusions and future work
Appendix A Vectorisation in MATLAB
Bibliography