Optimal A Priori Error Estimates for an Elliptic Problem with Dirac Right-Hand Side

Köppl, Tobias; Wohlmuth, Barbara

doi:10.1137/130927619

Cited by 42 publications

(48 citation statements)

References 21 publications

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“…This example shows that the convergence far from the singularity is faster, as the order of convergence in this case is 2 (see [21]). The difference between the convergence rates for the L 2 -norms on and 0 , led us to make the conjecture that the preponderant part of the error is concentrated around the singularity, as can be seen in Figures 4-7, which show the distribution of the error for 1/h 10, 15, 20, and 30.…”

Section: Concentration Of the Error Around The Singularitymentioning

confidence: 78%

“…They are also valid on more general domains as they include all convex polygones. Conversely, we prove a corollary to the result by Köppl and Wohlmuth [21] that gives optimal convergence in the H 1 -norm of the numerical approximation of problem (P δ ) in dimension 2 for Lagrange finite elements of degree k ≥ 2. In the case of Lagrange elements of degree 1 this result is quasioptimal.…”

Section: (P δ )mentioning

confidence: 81%

“…Proof In the particular case of Lagrange finite elements, Köppl and Wohlmuth proved in [21] the following convergence in the L 2 -norm of a subdomain which does not contain x 0 :…”

Section: Theorem 2 Consider a Bounded Open Convex Polygonal Domain ⊂ Rmentioning

confidence: 99%

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Local error estimates of the finite element method for an elliptic problem with a Dirac source term

Bertoluzza

Decoene

Lacouture

et al. 2017

Numerical Methods Partial

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The solutions of elliptic problems with a Dirac measure righthand side are not H 1 in dimension d ∈ {2, 3} and therefore the convergence of the finite element solutions is suboptimal in the L 2 -norm. In this article, we address the numerical analysis of the finite element method for the Laplace equation with Dirac source term: we consider, in dimension 3, the Dirac measure along a curve and, in dimension 2, the punctual Dirac measure. The study of this problem is motivated by the use of the Dirac measure as a reduced model in physical problems, for which high accuracy of the finite element method at the singularity is not required. We show a quasioptimal convergence in the H s -norm, for s ≥ 1 on subdomains which exclude the singularity; in the particular case of Lagrange finite elements, an optimal convergence in H 1 -norm is shown on a family of quasiuniform meshes. Our results are obtained using local Nitsche and Schatz-type error estimates, a weak version of Aubin-Nitsche duality lemma and a discrete inf-sup condition. These theoretical results are confirmed by numerical illustrations.

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Section: Concentration Of the Error Around The Singularitymentioning

confidence: 78%

Section: (P δ )mentioning

confidence: 81%

See 1 more Smart Citation

Local error estimates of the finite element method for an elliptic problem with a Dirac source term

Bertoluzza

Decoene

Lacouture

et al. 2017

Numerical Methods Partial

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“…While [19] considers this problem on a macroscopic scale, we model it on the micro-scale, where individual blood vessels can be resolved as part of our 1d domain. (4) as preconditioner with S approximated using (9).…”

Section: Discrete Preconditioner For the Coupled Problemmentioning

confidence: 99%

“…Another approach to the analysis of (2) has recently been suggested in the numerical study [2]. Building on the analysis of [9] for the elliptic problem with a 0 dimensional Dirac right-hand side, the wellposedness of the problem was shown with trial spaces W 1,p (Ω), p = 3 − d 2 and test spaces W 1,q (Ω), p −1 + q −1 = 1, and quasi-optimal error estimates for FEM shown in the norms which excluded a fixed neighborhood of Γ of radius R.…”

Section: Introductionmentioning

confidence: 99%

Sub-voxel Perfusion Modeling in Terms of Coupled 3d-1d Problem

Holter

Kuchta

Mardal

2019

Lecture Notes in Computational Science and Engineering

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A computational model for microcirculation including Fahraeus‐Lindqvist effect, plasma skimming and fluid exchange with the tissue interstitium

Possenti

Gregorio

Gerosa

et al. 2018

Numer Methods Biomed Eng

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We present a two phase model for microcirculation that describes the interaction of plasma with red blood cells. The model takes into account of typical effects characterizing the microcirculation, such as the Fahraeus-Lindqvist effect and plasma skimming. Besides these features, the model describes the interaction of capillaries with the surrounding tissue. More precisely, the model accounts for the interaction of capillary transmural flow with the surrounding interstitial pressure. Furthermore, the capillaries are represented as one-dimensional channels with arbitrary, possibly curved configuration. The latter two features rely on the unique ability of the model to account for variations of flow rate and pressure along the axis of the capillary, according to a local differential formulation of mass and momentum conservation. Indeed, the model stands on a solid mathematical foundation, which is also addressed in this work. In particular, we present the model derivation, the variational formulation and its approximation using the finite element method. Finally, we conclude the work with a comparative computational study of the importance of the Fahraeus-Lindqvist, plasma skimming and capillary leakage effects on the distribution of flow in a microvascular network.

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Optimal A Priori Error Estimates for an Elliptic Problem with Dirac Right-Hand Side

Cited by 42 publications

References 21 publications

Local error estimates of the finite element method for an elliptic problem with a Dirac source term

Local error estimates of the finite element method for an elliptic problem with a Dirac source term

Sub-voxel Perfusion Modeling in Terms of Coupled 3d-1d Problem

A computational model for microcirculation including Fahraeus‐Lindqvist effect, plasma skimming and fluid exchange with the tissue interstitium

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