Splitting method for elliptic equations with line sources

Gjerde, Ingeborg G.; Kumar, Kundan; Nordbotten, Jan Martin; Wohlmuth, Barbara

doi:10.48550/arxiv.1810.12979

Cited by 5 publications

(11 citation statements)

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“…Discussion of other discretization schemes can be found in [7], again with the restriction of two-dimensional grids. Enhancements include, among others, three-dimensional slanted wells [8,9], Green's functions for the computation of well indices [10,11,12], or the singularity subtraction method to obtain smooth solutions in the near-well region [13,14].…”

Section: Introductionmentioning

confidence: 99%

A new and consistent well model for one-phase flow in anisotropic porous media using a distributed source model

Koch

Helmig

Schneider

2020

Journal of Computational Physics

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A new well model for one-phase flow in anisotropic porous media is introduced, where the mass exchange between well and a porous medium is modeled by spatially distributed source terms over a small neighborhood region. To this end, we first present a compact derivation of the exact analytical solution for an arbitrarily oriented, infinite well cylinder in an infinite porous medium with anisotropic permeability tensor in R 3 , for constant well pressure and a given injection rate, using a conformal map. The analytical solution motivates the choice of a kernel function to distribute the sources. The presented model is independent from the discretization method and the choice of computational grids. In numerical experiments, the new well model is shown to be consistent and robust with respect to rotation of the well axis, rotation of the permeability tensor, and different anisotropy ratios. Finally, a comparison with a Peacemantype well model suggests that the new scheme leads to an increased accuracy for injection (and production) rates for arbitrarily-oriented pressure-controlled wells.

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Section: Introductionmentioning

confidence: 99%

A new and consistent well model for one-phase flow in anisotropic porous media using a distributed source model

Koch

Helmig

Schneider

2020

Journal of Computational Physics

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“…As we will see, the analysis raises questions regarding the approximation properties of q. For this reason, we extend our work from [19] to show that with some assumptions on the problem parameters, the solution admits a splitting of the type u = u s + u r , q = q s + q r , (1. 3) where u s and q s denote explicitly known terms capturing the solution singularities, and u r and q r denote higher regularity remainder terms.…”

Section: Introductionmentioning

confidence: 92%

“…With the splitting in hand, we then formulate a solution strategy in which only u r and q r are approximated using a mixed finite element method. In contrast to the development in [19], this has the advantage of providing a locally mass conservative approximation. The full solution pair (u, q) can then be reconstructed using (1.3).…”

Section: Introductionmentioning

confidence: 99%

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A Mixed Approach to the Poisson Problem with Line Sources

Gjerde¹,

Kumar²,

Nordbotten³

2019

Preprint

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In this work we consider the primal mixed variational formulation of the Poisson equation with a line source. The analysis and approximation of this problem is non-standard as the line source causes the solutions to be singular. We start by showing that this problem admits a solution in appropriately weighted Sobolev spaces. Next, we show that given some assumptions on the problem parameters, the solution admits a splitting into higher and lower regularity terms. The lower regularity terms are here explicitly known and capture the solution singularities. The higher regularity terms, meanwhile, are defined as the solution of its own mixed Poisson equation. With the solution splitting in hand, we then define a singularity removal based mixed finite element method in which only the higher regularity are approximated numerically. This method yields a significant improvement in the convergence rate when compared to approximating the full solution.In particular, we show that the singularity removal based method yields optimal convergence rates for lowest order Raviart-Thomas and discontinuous Lagrange elements.

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“…The modelling framework will entail Dirac Delta functions (distributions), where these pulse-like forces will lead to singularities of the solution in terms of a lower (local) degree of regularity, even such that the solution no longer falls within the finite-element space in which one looks for the solution. Some of the issues have been treated in [12], [13] and [14], regarding well-posedness and finite-element solutions. The treatment of momentum using point forces that we consider in the current paper was developed in [15], [7] and [8].…”

Section: Introductionmentioning

confidence: 99%

Numerical Methods to Compute Stresses and Displacements from Cellular Forces: Application to the Contraction of Tissue

Peng,

Vermolen

2020

Preprint

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We consider a mathematical model for wound contraction, which is based on solving a momentum balance under the assumptions of isotropy, homogeneity, Hooke's Law, infinitesimal strain theory and point forces exerted by cells. However, point forces, described by Dirac Delta distributions lead to a singular solution, which in many cases may cause trouble to finite element methods due to a low degree of regularity. Hence, we consider several alternatives to address point forces, that is, whether to treat the region covered by the cells that exert forces as part of the computational domain or as 'holes' in the computational domain. The formalisms develop into the immersed boundary approach and the 'hole approach', respectively. Consistency between these approaches is verified in a theoretical setting, but also confirmed computationally. However, the 'hole approach' is much more expensive and complicated for its need of mesh adaptation in the case of migrating cells while it increases the numerical accuracy, which makes it hard to adapt to the multi-cell model. Therefore, for multiple cells, we consider the polygon that is used to approximate the boundary of cells that exert contractile forces. It is found that a low degree of polygons, in particular triangular or square shaped cell boundaries, already give acceptable results in engineering precision, so that it is suitable for the situation with a large amount of cells in the computational domain.

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Splitting method for elliptic equations with line sources

Cited by 5 publications

References 0 publications

A new and consistent well model for one-phase flow in anisotropic porous media using a distributed source model

A new and consistent well model for one-phase flow in anisotropic porous media using a distributed source model

A Mixed Approach to the Poisson Problem with Line Sources

Numerical Methods to Compute Stresses and Displacements from Cellular Forces: Application to the Contraction of Tissue

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