BDDC for mixed‐hybrid formulation of flow in porous media with combined mesh dimensions

Abstract

SUMMARYWe extend the Balancing Domain Decomposition by Constraints (BDDC) method to flows in porous media discretised by mixed-hybrid finite elements with combined mesh dimensions. Such discretisations appear when major geological fractures are modelled by 1D or 2D elements inside three-dimensional domains. In this set-up, the global problem as well as the substructure problems have a symmetric saddle-point structure, containing a 'penalty' block due to the combination of meshes. We show that the problem can be reduced by means of iterative substructuring to an interface problem, which is symmetric and positive definite. The interface problem can thus be solved by conjugate gradients with the BDDC method as a preconditioner. A parallel implementation of this algorithm is incorporated into an existing software package for subsurface flow simulations. We study the performance of the iterative solver on several academic and real-world problems. Numerical experiments illustrate its efficiency and scalability. Preprint version.

Generation of Transport Paths in Fractured Porous Media

“…The procedure was originally invented for and tested on the base of Flow123d simulator [4,5,15] results. Flow123d is a software for a simulation of a water flow, reactive solute transport and heat transfer in a heterogeneous porous and fractured medium.…”

“…The proposed procedure is designed for the computation of the functional based on outputs of the Flow123d simulation software [4,5,15].…”

PCBDDC: A Class of Robust Dual-Primal Methods in PETSc

“…The algorithm also has been applied to a variety of PDEs including advection-diffusion [84], incompressible Stokes [55], porous media flows [82,83,73], almost incompressible elasticity [67], Reissner-Mindlin plates [11,53], and Naghdi shells [12]. The FETI-DP method has also been extensively studied; among others, we cite the studies on the Stokes problem [40], contact problems [6], indefinite complex problems [30,31], mortar discretizations [39], and biomechanics [49].…”

Preconditioning of iterative methods ‐ theory and applications

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The collection of five papers [2][3][4][5][6] published in this issue of Numerical Linear Algebra with Applications originates from the talks delivered at the conference Preconditioning of Iterative MethodsTheory and Applications (PIM 2013), organized in honor of Professor Ivo Marek on the occasion of his 80th birthday and represents a tribute to his work and influence in the numerical linear algebra community, compare with [1].We provide a short summary of the presented papers in order of their submission.

Šístek et al[2] deal with the solution of saddle point systems arising from the mixed-hybrid finite element discretization of flow in porous media including highly permeable fractures. The fractures are modeled by 1D or 2D elements inside three-dimensional domains.

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“…The collection of five papers [2][3][4][5][6] published in this issue of Numerical Linear Algebra with Applications originates from the talks delivered at the conference Preconditioning of Iterative MethodsTheory and Applications (PIM 2013), organized in honor of Professor Ivo Marek on the occasion of his 80th birthday and represents a tribute to his work and influence in the numerical linear algebra community, compare with [1].…”