Fast and Accuracy-Preserving Domain Decomposition Methods for Reduced Fracture Models with Nonconforming Time Grids

Huynh, Phuoc-Toan; Cao, Yanzhao; Hoang, Thi-Thao-Phuong

doi:10.1007/s10915-023-02251-0

J Sci Comput

2023

DOI: 10.1007/s10915-023-02251-0

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Fast and Accuracy-Preserving Domain Decomposition Methods for Reduced Fracture Models with Nonconforming Time Grids

Phuoc-Toan Huynh

Yanzhao Cao

Thi-Thao-Phuong Hoang

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2024

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Cited by 2 publications

References 35 publications

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A space-time mixed finite element method for reduced fracture flow models on nonmatching grids

Hoang,

Yotov

2024

Math. Comp.

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This paper is concerned with the numerical solution of the flow problem in a fractured porous medium where the fracture is treated as a lower dimensional object embedded in the rock matrix. We consider a space-time mixed variational formulation of such a reduced fracture model with mixed finite element approximations in space and discontinuous Galerkin discretization in time. Different spatial and temporal grids are used in the subdomains and in the fracture to adapt to the heterogeneity of the problem. Analysis of the numerical scheme, including well-posedness of the discrete problem, stability and a priori error estimates, is presented. Using substructuring techniques, the coupled subdomain and fracture system is reduced to a space-time interface problem which is solved iteratively by GMRES. Each GMRES iteration involves solution of time-dependent problems in the subdomains using the method of lines with local spatial and temporal discretizations. The convergence of GMRES is proved by using the field-of-values analysis and the properties of the discrete space-time interface operator. Numerical experiments are carried out to illustrate the performance of the proposed iterative algorithm and the accuracy of the numerical solution.

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A space-time mixed finite element method for reduced fracture flow models on nonmatching grids

Hoang,

Yotov

2024

Math. Comp.

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Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media

Cao,

Hoang,

Huynh

2024

IMA Journal of Numerical Analysis

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The objective of this paper is to develop efficient numerical algorithms for the linear advection-diffusion equation in fractured porous media. A reduced fracture model is considered where the fractures are treated as interfaces between subdomains and the interactions between the fractures and the surrounding porous medium are taken into account. The model is discretized by a backward Euler upwind-mixed hybrid finite element method in which the flux variable represents both the advective and diffusive fluxes. The existence, uniqueness, as well as optimal error estimates in both space and time for the fully discrete coupled problem are established. Moreover, to facilitate different time steps in the fracture-interface and the subdomains, global-in-time, nonoverlapping domain decomposition is utilized to derive two implicit iterative solvers for the discrete problem. The first method is based on the time-dependent Steklov–Poincaré operator, while the second one employs the optimized Schwarz waveform relaxation (OSWR) approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively with possibly variable time step sizes. The convergence of the OSWR-based method with conforming time grids is also proved. Finally, numerical results in two dimensions are presented to verify the optimal order of convergence of the monolithic solver and to illustrate the performance of the two decoupled schemes with local time-stepping on problems of high Péclet numbers.

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Fast and Accuracy-Preserving Domain Decomposition Methods for Reduced Fracture Models with Nonconforming Time Grids

Cited by 2 publications

References 35 publications

A space-time mixed finite element method for reduced fracture flow models on nonmatching grids

A space-time mixed finite element method for reduced fracture flow models on nonmatching grids

Monolithic and local time-stepping decoupled algorithms for transport problems in fractured porous media

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