| 1. |
- Boon, Wietse M., et al.
(författare)
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Convergence of a tpfa finite volume scheme for mixed-dimensional flow problems
- 2020
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Ingår i: Springer Proceedings in Mathematics and Statistics. - Cham : Springer. ; , s. 435-444
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Konferensbidrag (refereegranskat)abstract
- A two-point flux approximation (TPFA) finite volume method is considered for mixed-dimensional fracture flow problems. Its construction is based on applying a face-based quadrature rule to a conforming, mixed finite element scheme of lowest order. A concise argument shows linear convergence in theory, which we confirm in practice by a numerical experiment.
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| 2. |
- Boon, Wietse M., et al.
(författare)
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Functional analysis and exterior calculus on mixed-dimensional geometries
- 2020
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Ingår i: Annali di Matematica Pura ed Applicata. - : Springer. - 0373-3114 .- 1618-1891.
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Tidskriftsartikel (refereegranskat)abstract
- We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of d-dimensional manifolds, structured hierarchically so that each d-dimensional manifold is contained in the boundary of one or more d+ 1 -dimensional manifolds. On any given d-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete differential operators (jumps) normal to the manifold. The combined action of these operators leads to the notion of a semi-discrete differential operator coupling manifolds of different dimensions. We refer to the resulting systems of equations as mixed-dimensional, which have become a popular modeling technique for physical applications including fractured and composite materials. We establish analytical tools in the mixed-dimensional setting, including suitable inner products, differential and codifferential operators, Poincaré lemma, and Poincaré–Friedrichs inequality. The manuscript is concluded by defining the mixed-dimensional minimization problem corresponding to the Hodge Laplacian, and we show that this minimization problem is well-posed.
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| 3. |
- Boon, Wietse M.
(författare)
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A parameter-robust iterative method for Stokes-Darcy problems retaining local mass conservation
- 2020
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Ingår i: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 1290-3841. ; 54:6, s. 2045-2067
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Tidskriftsartikel (refereegranskat)abstract
- We consider a coupled model of free-flow and porous medium flow, governed by stationary Stokes and Darcy flow, respectively. The coupling between the two systems is enforced by introducing a single variable representing the normal flux across the interface. The problem is reduced to a system concerning only the interface flux variable, which is shown to be well-posed in appropriately weighted norms. An iterative solution scheme is then proposed to solve the reduced problem such that mass is conserved at each iteration. By introducing a preconditioner based on the weighted norms from the analysis, the performance of the iterative scheme is shown to be robust with respect to material and discretization parameters. By construction, the scheme is applicable to a wide range of locally conservative discretization schemes and we consider explicit examples in the framework of Mixed Finite Element methods. Finally, the theoretical results are confirmed with the use of numerical experiments.
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| 4. |
- Budisa, Ana, et al.
(författare)
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Mixed-dimensional auxiliary space preconditioners
- 2020
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 42:5, s. A3367-A3396
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Tidskriftsartikel (refereegranskat)abstract
- 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 T. Numer. Anal., 45 (2007), pp. 2483-2509]. Numerical experiments show the performance of this preconditioner in the context of flow in fractured porous media.
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