Preconditioning Techniques for the Numerical Solution of Flow in Fractured Porous Media

Antonietti, Paola F.; Ponti, Jacopo De; Formaggia, Luca; Scotti, Anna

doi:10.1007/s10915-020-01372-0

Cited by 5 publications

(3 citation statements)

References 55 publications

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“…However, we see from Theorem 5.9 that the condition number of the DD system depends on h and \alpha \gamma , which in turn influences the number of iterations of the iterative solver. To retain robustness, we can use a preconditioner [7,18] or a coarse mortar space that is compensated by taking higher-order mortars [9,50].…”

Section: Lemma 55 (Boundedness On a \Gamma )mentioning

confidence: 99%

Robust Linear Domain Decomposition Schemes for Reduced Nonlinear Fracture Flow Models

Ahmed¹,

Fumagalli²,

Budiša³

et al. 2021

SIAM J. Numer. Anal.

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\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . In this work, we consider compressible single-phase flow problems in a porous medium containing a fracture. In the fracture, a nonlinear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we reformulate the global problem into a nonlinear interface problem. We then introduce two new algorithms that are able to efficiently handle the nonlinearity and the coupling between the fracture and the matrix, both based on linearization by the so-called L-scheme. The first algorithm, named MoLDD, uses the L-scheme to resolve for the nonlinearity, requiring at each iteration to solve the dimensional coupling via a domain decomposition approach. The second algorithm, called ItLDD, uses a sequential approach in which the dimensional coupling is part of the linearization iterations. For both algorithms, the computations are reduced only to the fracture by precomputing, in an offline phase, a multiscale flux basis (the linear Robin-to-Neumann codimensional map), that represent the flux exchange between the fracture and the matrix. We present extensive theoretical findings X and in particular, t. The stability and the convergence of both schemes are obtained, where user-given parameters are optimized to minimize the number of iterations. Examples on two important fracture models are computed with the library PorePy and agree with the developed theory.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . porous medium, reduced fracture models, generalized Forchheimer's laws, mortar mixed finite element, multiscale flux basis, nonlinear interface problem, nonoverlapping domain decomposition, L-scheme \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 76S05, 65N30, 65N12 \bfD \bfO \bfI .

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Section: Lemma 55 (Boundedness On a \Gamma )mentioning

confidence: 99%

Robust Linear Domain Decomposition Schemes for Reduced Nonlinear Fracture Flow Models

Ahmed¹,

Fumagalli²,

Budiša³

et al. 2021

SIAM J. Numer. Anal.

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show abstract

“…Thus, the MFB framework is favourable for 1) highly heterogeneous parts of the media where subdomain solves are affected by heterogeneities, 2) those fractures affected by strong non-linearities, and 3) lower permeable or blocking fractures where a coarse mortar space can be used without sacrificing accuracy. Otherwise, a robust preconditioner [7,18] can be used in the Krylov method, as well as a coarse mortar space that is compensated by taking higher order mortars [9,55].…”

Section: Form the Multiscale Flux Basis For Subdomainmentioning

confidence: 99%

Robust linear domain decomposition schemes for reduced non-linear fracture flow models

Ahmed¹,

Fumagalli²,

Budiša³

et al. 2019

Preprint

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In this work, we consider compressible single-phase flow problems in a porous media containing a fracture. In the latter, a non-linear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we reformulate the global problem into a non-linear interface problem. We then introduce two new algorithms that are able to efficiently handle the non-linearity and the coupling between the fracture and the matrix, both based on linearization by the so-called L-scheme. The first algorithm, named MoLDD, uses the L-scheme to resolve the non-linearity, requiring at each iteration to solve the dimensional coupling via a domain decomposition approach. The second algorithm, called ItLDD, uses a sequential approach in which the dimensional coupling is part of the linearization iterations. For both algorithms, the computations are reduced only to the fracture by pre-computing, in an offline phase, a multiscale flux basis (the linear Robin-to-Neumann co-dimensional map), that represent the flux exchange between the fracture and the matrix. We present extensive theoretical findings and in particular, the stability and the convergence of both schemes are obtained, where usergiven parameters are optimized to minimise the number of iterations. Examples on two important fracture models are computed with the library PorePy and agree with the developed theory.

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“…Several approaches have recently been proposed in the context of preconditioners for fracture flow problems, in both an equidimensional setting [31] and mixeddimensional formulations [6,8,13]. Similar to [13], we design robust block preconditioners based on the well-posedness of the discrete PDEs using the framework developed in [26,27].…”

mentioning

confidence: 99%

Mixed-Dimensional Auxiliary Space Preconditioners

Budiša¹,

Boon²,

Hu³

2020

SIAM J. Sci. Comput.

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MIXED-DIMENSIONAL AUXILIARY SPACE PRECONDITIONERS \astANA BUDI \v SA \dagger , WIETSE M. BOON \ddagger , AND XIAOZHE HU \S \bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . This work introduces nodal auxiliary space preconditioners for discretizations of mixed-dimensional partial differential equations. We first consider the continuous setting and generalize the regular decomposition to this setting. With the use of conforming mixed finite element spaces, we then expand these results to the discrete case and obtain a decomposition in terms of nodal Lagrange elements. In turn, nodal preconditioners are proposed analogous to the auxiliary space preconditioners of Hiptmair and Xu [SIAM J. Numer. Anal., 45 (2007), pp. 2483--2509. Numerical experiments show the performance of this preconditioner in the context of flow in fractured porous media.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . finite element, mixed-dimensional, iterative method, auxiliary space preconditioning, algebraic multigrid method, fracture flow \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 65F08, 65N30, 65N55 \bfD \bfO \bfI .

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Preconditioning Techniques for the Numerical Solution of Flow in Fractured Porous Media

Cited by 5 publications

References 55 publications

Robust Linear Domain Decomposition Schemes for Reduced Nonlinear Fracture Flow Models

Robust Linear Domain Decomposition Schemes for Reduced Nonlinear Fracture Flow Models

Robust linear domain decomposition schemes for reduced non-linear fracture flow models

Mixed-Dimensional Auxiliary Space Preconditioners

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