A Local Error Estimate for the Poisson Equation with a Line Source Term

Köppl, Tobias; Vidotto, Ettore; Wohlmuth, Barbara

doi:10.1007/978-3-319-39929-4_40

Cited by 19 publications

(24 citation statements)

References 7 publications

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“…Seminal works featuring the formulation and analysis of bulk 3D with embedded 1D PDEs, due to D'Angelo and co-workers [12][13][14]30], have appeared much later than their applications. From the point of view of numerical approximation and analysis of such problems, besides the works by D'Angelo, we mention [4,6,[24][25][26][27][28]33]. More precisely, in [4,24,25] optimal a-priori error estimates for the finite element approximation of elliptic equations in with Dirac sources are addressed.…”

Section: Introductionmentioning

confidence: 99%

Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction

Laurino

Zunino

2019

ESAIM: M2AN

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Multiscale methods based on coupled partial differential equations defined on bulk and embedded manifolds are still poorly explored from the theoretical standpoint, although they are successfully used in applications, such as microcirculation and flow in perforated subsurface reservoirs. This work aims at shedding light on some theoretical aspects of a multiscale method consisting of coupled partial differential equations defined on one-dimensional domains embedded into three-dimensional ones. Mathematical issues arise because the dimensionality gap between the bulk and the inclusions is larger than one, that is the high dimensionality gap case. First, we show that such model derives from a system of fully three-dimensional equations, by the application of a topological model reduction approach. Secondly, we rigorously analyze the problem, showing that the averaging operators applied for the model reduction introduce a regularization effect that resolves the issues due to the singularity of solutions and to the ill-posedness of restriction operators. Then, we exploit the structure of the model reduction technique to analyze the modeling error. This study confirms that for infinitesimally small inclusions, the modeling error vanishes. Finally, we discretize the problem by means of the finite element method and we analyze the approximation and the model error by means of numerical experiments.

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Section: Introductionmentioning

confidence: 99%

Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction

Laurino

Zunino

2019

ESAIM: M2AN

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“…The difference to the presented coupling approach is that in the source term of the tissue problem, the Dirac measure is concentrated on the midlines of the vascular system. As a result singularities along the network midlines are introduced in the 3D pressure field [31,22]. Moreover, there are no estimates for the modeling errors arising from this type of coupling concepts.…”

Section: Fully-discrete 3d-1d Modelmentioning

confidence: 99%

“…By this, an expensive meshing of a three-dimensional (3D) blood vessel system is avoided and at the same time the hierarchical structure of the vascular system is maintained. However, elaborate concepts for coupling one-dimensional flow equations with a flow equation (Darcy equation) for a three-dimensional flow problem [10,31,30] have to be developed. This is done by constructing suitable source terms for both the Darcy equation in 3D and the 1D flow problems.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Models for Simulating Blood Flow in Microvascular Networks

Vidotto¹,

Koch²,

Köppl³

et al. 2019

Multiscale Model. Simul.

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In this paper, we are concerned with the simulation of blood flow in microvascular networks and the surrounding tissue. To reduce the computational complexity of this issue, the network structures are modeled by a one-dimensional graph, whose location in space is determined by the centerlines of the three-dimensional vessels. The surrounding tissue is considered as a homogeneous porous medium. Darcy's equation is used to simulate flow in the extra-vascular space, where the mass exchange with the blood vessels is accounted for by means of line source terms. However, this model reduction approach still causes high computational costs, in particular, when larger parts of an organ have to be simulated. This observation motivates the consideration of a further model reduction step. Thereby, we homogenize the fine scale structures of the microvascular networks resulting in a new hybrid approach modeling the fine scale structures as a heterogeneous porous medium and the flow in the larger vessels by one-dimensional flow equations. Both modeling approaches are compared with respect to mass fluxes and averaged pressures. The simulations have been performed on a microvascular network that has been extracted from the cortex of a rat brain.

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“…The herein presented fluid‐mechanical model is similar to the drug proliferation model described in Cattaneo and Zunino and introduced in Possenti et al It is derived here for the specific application of contrast agent perfusion in brain tissue. The mathematical background of such models is analyzed in several works …”

Section: Introductionmentioning

confidence: 99%

A multiscale subvoxel perfusion model to estimate diffusive capillary wall conductivity in multiple sclerosis lesions from perfusion MRI data

Koch

Flemisch

Helmig

et al. 2020

Numer Methods Biomed Eng

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We propose a new mathematical model to learn capillary leakage coefficients from dynamic susceptibility contrast MRI data. To this end, we derive an embedded mixed‐dimension flow and transport model for brain tissue perfusion on a subvoxel scale. This model is used to obtain the contrast agent concentration distribution in a single MRI voxel during a perfusion MRI sequence. We further present a magnetic resonance signal model for the considered sequence including a model for local susceptibility effects. This allows modeling MR signal‐time curves that can be compared with clinical MRI data. The proposed model can be used as a forward model in the inverse modeling problem of inferring model parameters such as the diffusive capillary wall conductivity. Acute multiple sclerosis lesions are associated with a breach in the integrity of the blood‐brain barrier. Applying the model to perfusion MR data of a patient with acute multiple sclerosis lesions, we conclude that diffusive capillary wall conductivity is a good indicator for characterizing activity of lesions, even if other patient‐specific model parameters are not well‐known.

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A Local Error Estimate for the Poisson Equation with a Line Source Term

Cited by 19 publications

References 7 publications

Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction

Derivation and analysis of coupled PDEs on manifolds with high dimensionality gap arising from topological model reduction

Hybrid Models for Simulating Blood Flow in Microvascular Networks

A multiscale subvoxel perfusion model to estimate diffusive capillary wall conductivity in multiple sclerosis lesions from perfusion MRI data

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