Mixed-Dimensional Auxiliary Space Preconditioners

Budiša, Ana; Boon, Wietse M.; Hu, Xiaozhe

doi:10.1137/19m1292618

Cited by 9 publications

(7 citation statements)

References 25 publications

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“…One way of resolving this is to use auxiliary space theory (see, for example, [22,25]). However, in our case, additional theory resulting from the mixed-dimensional exterior calculus in [8] is needed, which is the topic of our ongoing work [12]. However, in this paper, we consider an alternative formulation of the problem (2.22) and show the well-posedness with respect to a different weighted norm, which allows for a simpler robust preconditioner.…”

Section: Robust Preconditioners For Mixed-dimensional Modelmentioning

confidence: 99%

Block preconditioners for mixed-dimensional discretization of flow in fractured porous media

Budiša

2020

Comput Geosci

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In this paper, we are interested in an efficient numerical method for the mixed-dimensional approach to modeling single-phase flow in fractured porous media. The model introduces fractures and their intersections as lower-dimensional structures, and the mortar variable is used for flow coupling between the matrix and fractures. We consider a stable mixed finite element discretization of the problem, which results in a parameter-dependent linear system. For this, we develop block preconditioners based on the well-posedness of the discretization choice. The preconditioned iterative method demonstrates robustness with regard to discretization and physical parameters. The analytical results are verified on several examples of fracture network configurations, and notable results in reduction of number of iterations and computational time are obtained.

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Section: Robust Preconditioners For Mixed-dimensional Modelmentioning

confidence: 99%

Block preconditioners for mixed-dimensional discretization of flow in fractured porous media

Budiša

2020

Comput Geosci

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“…The mixed-dimensional structure was used from PorePy [18], with the grids created using the meshing software GMSH [14]. Finally, the mixed-dimensional curl operator was adapted from [11].…”

Section: These Matrices Allow Us To Compute the Operator Dmentioning

confidence: 99%

A Reduced Basis Method for Darcy flow systems that ensures local mass conservation by using exact discrete complexes

Boon¹,

Fumagalli²

2022

Preprint

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A solution technique is proposed for flows in porous media that guarantees local conservation of mass. We first compute a flux field to balance the mass source and then exploit exact co-chain complexes to generate a solenoidal correction. A reduced basis method based on proper orthogonal decomposition is employed to construct the correction and we show that mass balance is ensured regardless of the quality of the reduced basis approximation. The method is directly applicable to mixed finite and virtual element methods, among other structure-preserving discretization techniques, and we present the extension to Darcy flow in fractured porous media.Keywords Reduced basis method • exact discrete complex • mixed finite element method • virtual element method • fractured porous media

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“…In particular, the beam field usually results in non‐diagonally dominant (sub‐)matrices leading to the need for advanced preconditioning. In addition, the mixed‐dimensional nature of the problem calls for specially‐tailored preconditioning for the overall monolithic problem as analyzed in References 23–25. These challenges and open questions in the context of efficient linear solvers can be bypassed by choosing a strongly‐coupled staggered partitioned solution approach.…”

Section: Introductionmentioning

confidence: 99%

A fully coupled regularized mortar‐type finite element approach for embedding one‐dimensional fibers into three‐dimensional fluid flow

Hagmeyer,

Mayr,

Popp

2024

Numerical Meth Engineering

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SummaryThe present article proposes a partitioned Dirichlet‐Neumann algorithm, that allows to address unique challenges arising from a novel mixed‐dimensional coupling of very slender fibers embedded in fluid flow using a regularized mortar‐type finite element discretization. The fibers are modeled via one‐dimensional (1D) partial differential equations based on geometrically exact nonlinear beam theory, while the flow is described by the three‐dimensional (3D) incompressible Navier‐Stokes equations. The arising truly mixed‐dimensional 1D‐3D coupling scheme constitutes a novel approximate model and numerical strategy, that naturally necessitates specifically tailored solution schemes to ensure an accurate and efficient computational treatment. In particular, we present a strongly coupled partitioned solution algorithm based on a Quasi‐Newton method for applications involving fibers with high slenderness ratios that usually present a challenge with regard to the well‐known added mass effect. The influence of all employed algorithmic and numerical parameters, namely the applied acceleration technique, the employed constraint regularization parameter as well as shape functions, on efficiency and results of the solution procedure is studied through appropriate examples. Finally, the convergence of the two‐way coupled mixed‐dimensional problem solution under uniform mesh refinement is demonstrated, a comparison to a 3D reference solution is performed, and the method's capabilities in capturing flow phenomena at large geometric scale separation is illustrated by the example of a submersed vegetation canopy.

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Mixed-Dimensional Auxiliary Space Preconditioners

Cited by 9 publications

References 25 publications

Block preconditioners for mixed-dimensional discretization of flow in fractured porous media

Block preconditioners for mixed-dimensional discretization of flow in fractured porous media

A Reduced Basis Method for Darcy flow systems that ensures local mass conservation by using exact discrete complexes

A fully coupled regularized mortar‐type finite element approach for embedding one‐dimensional fibers into three‐dimensional fluid flow

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