A Global-in-time Domain Decomposition Method for the Coupled Nonlinear Stokes and Darcy Flows

Hoang, Thi-Thao-Phuong; Lee, Hyesuk

doi:10.1007/s10915-021-01422-1

Cited by 15 publications

(7 citation statements)

References 60 publications

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“…In practice, the ratio between the coarse and fine time steps is determined on a case-by-case basis -depending on the jumps in the coefficients as shown numerically in [31,34,30] and the next section. We also remark that global-in-time domain decomposition methods are implicit in time, thus large time steps can be used (without affecting the numerical stability) when the model parameters are small [35].…”

Section: Nonconforming Time Discretizationsmentioning

confidence: 99%

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Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

Hoang

2022

DCDS-S

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<p style='text-indent:20px;'>This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Peclét numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.</p>

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Section: Nonconforming Time Discretizationsmentioning

confidence: 99%

“…We will compute the terms involving (φ i,Γ,EP ) E∈E Γ h ,P ∈∂E in (38) by using the relation (25) between φ φ φ i,Γ and the solution of the subdomain problem ( 30)- (35). Taking v v v Γ to be the basis functions of Σ h,Γ in (25) and after some simple calculations, we obtain 2 −1 −1 2…”

mentioning

confidence: 99%

Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

Hoang

2022

DCDS-S

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“…There are basically two types of global-in-time DD methods. The first type is based on the physical transmission conditions, for example, the Dirichlet-Neumann and Neumann-Neumann waveform relaxation methods [17,18,26,29,31]. The second type is based on more general transmission conditions such as Robin or Ventcel [40] conditions.…”

Section: Introductionmentioning

confidence: 99%

Fast and accurate domain decomposition methods for reduced fracture models with nonconforming time grids

Huynh¹,

Cao²,

Hoang³

2022

Preprint

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This paper is concerned with the numerical solution of compressible fluid flow in a fractured porous medium. The fracture represents a fast pathway (i.e., with high permeability) and is modeled as a hypersurface embedded in the porous medium. We aim to develop fast-convergent and accurate global-in-time domain decomposition (DD) methods for such a reduced fracture model, in which smaller time step sizes in the fracture can be coupled with larger time step sizes in the subdomains. Using the pressure continuity equation and the tangential PDEs in the fracture-interface as transmission conditions, three different DD formulations are derived; each method leads to a space-time interface problem which is solved iteratively and globally in time. Efficient preconditioners are designed to accelerate the convergence of the iterative methods while preserving the accuracy in time with nonconforming grids. Numerical results for two-dimensional problems with non-immersed and partially immersed fractures are presented to show the improved performance of the proposed methods.

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“…Parallelism in time has also been explored, such as the Parareal algorithm [27,42] and multigrid in time [20,26]. Local time stepping techniques have been developed for multiphysics systems coupled through interface conditions, e.g., for the Stokes-Darcy system [35,47]. Finally, we mention some earlier works on space-time methods for parabolic problems coupling mixed finite element discretizations in space with DG in time on a single domain.…”

Section: Introductionmentioning

confidence: 99%

A space-time multiscale mortar mixed finite element method for parabolic equations

Jayadharan¹,

Kern²,

Vohralı́k³

et al. 2021

Preprint

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We develop a space-time mortar mixed finite element method for parabolic problems. The domain is decomposed into a union of subdomains discretized with non-matching spatial grids and asynchronous time steps. The method is based on a space-time variational formulation that couples mixed finite elements in space with discontinuous Galerkin in time. Continuity of flux (mass conservation) across space-time interfaces is imposed via a coarse-scale space-time mortar variable that approximates the primary variable. Uniqueness, existence, and stability, as well as a priori error estimates for the spatial and temporal errors are established. A space-time non-overlapping domain decomposition method is developed that reduces the global problem to a space-time coarse-scale mortar interface problem. Each interface iteration involves solving in parallel space-time subdomain problems. The spectral properties of the interface operator and the convergence of the interface iteration are analyzed. Numerical experiments are provided that illustrate the theoretical results and the flexibility of the method for modeling problems with features that are localized in space and time.

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A Global-in-time Domain Decomposition Method for the Coupled Nonlinear Stokes and Darcy Flows

Cited by 15 publications

References 60 publications

Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

Fast and accurate domain decomposition methods for reduced fracture models with nonconforming time grids

A space-time multiscale mortar mixed finite element method for parabolic equations

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