A posteriori error estimates and stopping criteria for solvers using the domain decomposition method and with local time stepping

Abstract : This work contributes to the developpement of a posteriori error estimates and stopping criteria for domain decomposition methods with optimized Robin transmission conditions on the interface between subdomains. We study several problems. First, we tackle the steady diffusion equation using the mixed finite element subdomain discretization. Then the heat equation using the mixed finite element method in space and the discontinuous Galerkin scheme of lowest order in time is investigated. For the heat equation, a global-in-time domain decomposition method is used for both conforming and nonconforming time grids allowing for different time steps in different subdomains. This work is then extended to a two-phase flow model using a finite volume scheme in space. For each model, the multidomain formulation can be rewritten as an interface problem which is solved iteratively. Here at each iteration, local subdomain problems are solved, and information is then transferred to the neighboring subdomains. For unsteady problems, the subdomain problems are time-dependent and information is transferred via a space-time interface. The aim of this work is to bound the error between the exact solution and the approximate solution at each iteration of the domain decomposition algorithm. Different error components, such as the domain decomposition error, are identified in order to define efficient stopping criteria for the domain decomposition algorithm. More precisely, for the steady diffusion problem, the error of the domain decomposition method and that of the discretization in space are estimated separately. In addition, the time error for the unsteady problems is identified. Our a posteriori estimates are based on the reconstruction techniques for pressures and fluxes respectively in the spaces H1 and H(div). For the fluxes, local Neumann problems in bands arround the interfaces extracted from the subdomains are solved. Consequently, an effective criterion to stop the domain decomposition iterations is developed. Numerical experiments, both academic and more realistic with industrial data, are shown to illustrate the efficiency of these techniques. In particular, different time steps in different subdomains for the industrial example are used.
Document type :
Theses
Complete list of metadatas

Cited literature [148 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-01672497
Contributor : Abes Star <>
Submitted on : Thursday, December 28, 2017 - 1:01:37 AM
Last modification on : Thursday, February 7, 2019 - 1:33:15 AM

File

2017PA066098.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-01672497, version 2

Citation

Sarah Ali Hassan. A posteriori error estimates and stopping criteria for solvers using the domain decomposition method and with local time stepping. Numerical Analysis [math.NA]. Sorbonne Université / Université Pierre et Marie Curie - Paris VI, 2017. English. ⟨NNT : 2017PA066098⟩. ⟨tel-01672497v2⟩

Share

Metrics

Record views

415

Files downloads

247