Free Access
Issue
ESAIM: M2AN
Volume 53, Number 6, November-December 2019
Page(s) 2047 - 2080
DOI https://doi.org/10.1051/m2an/2019042
Published online 10 December 2019
  1. S. Alinhac and P. Gérard, Pseudo-differential operators and the Nash-Moser theorem. In: Vol. 82 of Graduate Studies in Mathematics. Translated from the 1991 French original. American Mathematical Society, Providence, RI (2007). [CrossRef] [Google Scholar]
  2. M. Arioli and M. Benzi, A finite element method for quantum graphs. IMA J. Numer. Anal. 38 (2018) 1119–1163. [CrossRef] [Google Scholar]
  3. G. Berkolaiko, R. Carlson, S.A. Fulling and P. Kuchment, Quantum graphs and their applications. In: Vol. 415 of Contemporary Mathematics. American Mathematical Society, Providence, RI (2006) 97–120. [Google Scholar]
  4. S. Bertoluzza, A. Decoene, L. Lacouture and S. Martin, Local error estimates of the finite element method for an elliptic problem with a Dirac source term. Numer. Method. Part. Differ. Equ. 34 (2018) 97–120. [CrossRef] [Google Scholar]
  5. T.R. Blake and J.F. Gross, Analysis of coupled intra- and extraluminal flows for single and multiple capillaries. Math. Biosci. 59 (1982) 173–206. [Google Scholar]
  6. W. Boon, J. Nordbotten and J. Vatne, Functional analysis and exterior calculus on mixed-dimensional geometries, Technical Report, Cornell University Library. Preprint arXiv:1710.00556v3 (2018). [Google Scholar]
  7. M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 1 (2003) 221–238. [Google Scholar]
  8. S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal. 41 (2003) 306–324. [Google Scholar]
  9. L. Cattaneo and P. Zunino, A computational model of drug delivery through microcirculation to compare different tumor treatments. Int. J. Numer. Method. Biomed. Eng. 30 (2014) 1347–1371. [CrossRef] [PubMed] [Google Scholar]
  10. L. Cattaneo and P. Zunino, Computational models for fluid exchange between microcirculation and tissue interstitium. Netw. Heterog. Media 9 (2014) 135–159. [CrossRef] [Google Scholar]
  11. D. Cerroni, F. Laurino and P. Zunino, Mathematical analysis, finite element approximation and numerical solvers for the interaction of 3D reservoirs with 1D wells. GEM – Int. J. Geomath. 10 (2019) 4. [CrossRef] [Google Scholar]
  12. C. D’Angelo, Multi scale modelling of metabolism and transport phenomena in living tissues. Ph.D. thesis, EPFL, Lausanne (2007). [Google Scholar]
  13. C. D’Angelo, Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces: applications to one- and three-dimensional coupled problems. SIAM J. Numer. Anal. 50 (2012) 194–215. [Google Scholar]
  14. C. D’Angelo and A. Quarteroni, the coupling of 1D and 3D diffusion-reaction equations. Application to tissue perfusion problems. Math. Model. Method. Appl. Sci. 18 (2008) 1481–1504. [CrossRef] [MathSciNet] [Google Scholar]
  15. A. Ern and J.-L. Guermond, Theory and practice of finite elements. In: Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). [CrossRef] [Google Scholar]
  16. J.F. Bonder and J.D. Rossi, Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Commun. Pure Appl. Anal. 1 (2002) 359–378. [CrossRef] [Google Scholar]
  17. G.J. Fleischman, T.W. Secomb and J.F. Gross, The interaction of extravascular pressure fields and fluid exchange in capillary networks. Math. Biosci. 82 (1986) 141–151. [Google Scholar]
  18. G.J. Flieschman, T.W. Secomb and J.F. Gross, Effect of extravascular pressure gradients on capillary fluid exchange. Math. Biosci. 81 (1986) 145–164. [Google Scholar]
  19. I. Gansca, W.F. Bronsvoort, G. Coman and L. Tambulea, Self-intersection avoidance and integral properties of generalized cylinders. Comput. Aided Geom. Design 19 (2002) 695–707. [CrossRef] [Google Scholar]
  20. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. In: Classics in Mathematics. Reprint of the 1998 edition. Springer-Verlag, Berlin (2001). [Google Scholar]
  21. I. Gjerde, K. Kumar, J.M. Nordbotten and B. Wohlmuth, Splitting method for elliptic equations with line sources. ESAIM: M2AN 53 (2019) 1715–1739. [CrossRef] [EDP Sciences] [Google Scholar]
  22. W. Gong, G. Wang and N. Yan, Approximations of elliptic optimal control problems with controls acting on a lower dimensional manifold. SIAM J. Control Optim. 52 (2014) 2008–2035. [Google Scholar]
  23. T. Koch, K. Heck, N. Schrder, H. Class and R. Helmig, A new simulation framework for soilroot interaction, evaporation, root growth, and solute transport. Vadose Zone J. 17 (2018) 0210. [Google Scholar]
  24. T. Köppl, E. Vidotto and B. Wohlmuth, A local error estimate for the Poisson equation with a line source term. In: Numerical Mathematics and Advanced Applications ENUMATH 2015. Springer (2016) 421–429. [CrossRef] [Google Scholar]
  25. T. Köppl and B. Wohlmuth, Optimal a priori error estimates for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal. 52 (2014) 1753–1769. [Google Scholar]
  26. T. Köppl, E. Vidotto, B. Wohlmuth and P. Zunino, Mathematical modeling, analysis and numerical approximation of second-order elliptic problems with inclusions. Math. Model. Method. Appl. Sci. 28 (2018) 953–978. [CrossRef] [Google Scholar]
  27. M. Kuchta, M. Nordaas, J.C.G. Verschaeve, M. Mortensen and K.-A. Mardal, Preconditioners for saddle point systems with trace constraints coupling 2D and 1D domains. SIAM J. Sci. Comput. 38 (2016) B962–B987. [Google Scholar]
  28. M. Kuchta, K.-A. Mardal and M. Mortensen, Preconditioning trace coupled 3D–1D systems using fractional Laplacian. Numer. Method. Partial Differ. Equ. 35 (2019) 375–393. [CrossRef] [Google Scholar]
  29. J.R. Kuttler and V.G. Sigillito, An inequality of a Stekloff eigenvalue by the method of defect. Proc. Am. Math. Soc. 20 (1969) 357–360. [Google Scholar]
  30. M. Lesinigo, C. D’Angelo and A. Quarteroni, A multiscale Darcy-Brinkman model for fluid flow in fractured porous media. Numer. Math. 117 (2011) 717–752. [Google Scholar]
  31. M. Nabil, P. Decuzzi and P. Zunino, Modelling mass and heat transfer in nano-based cancer hyperthermia. R. Soc. Open Sci. 2 (2015) 150447. [CrossRef] [PubMed] [Google Scholar]
  32. M. Nabil and P. Zunino, A computational study of cancer hyperthermia based on vascular magnetic nanoconstructs. R. Soc. Open Sci. 3 (2016) 160287. [CrossRef] [PubMed] [Google Scholar]
  33. D. Notaro, L. Cattaneo, L. Formaggia, A. Scotti and P. Zunino, A Mixed Finite Element Method for Modeling the Fluid Exchange Between Microcirculation and Tissue Interstitium. Springer International Publishing (2016) 3–25. [Google Scholar]
  34. L.E. Payne and H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5 (1960) 286–292. [Google Scholar]
  35. D.W. Peaceman, Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability. Soc. Petrol. Eng. J. 23 (1983) 531–543. [CrossRef] [Google Scholar]
  36. D.W. Peaceman, Interpretation of well-block pressures in numerical reservoir simulation. Soc. Petrol. Eng. AIME J. 18 (1978) 183–194. [CrossRef] [Google Scholar]
  37. L. Possenti, G. Casagrande, S. Di Gregorio, P. Zunino and M.L. Costantino, Numerical simulations of the microvascular fluid balance with a non-linear model of the lymphatic system. Microvasc. Res. 122 (2019) 101–110. [Google Scholar]
  38. L. Possenti, S. di Gregorio, F.M. Gerosa, G. Raimondi, G. Casagrande, M.L. Costantino and P. Zunino, A computational model for microcirculation including Fahraeus-Lindqvist effect, plasma skimming and fluid exchange with the tissue interstitium. Int. J. Numer. Method. Biomed. Eng. 35 (2019) e3165. [CrossRef] [PubMed] [Google Scholar]
  39. A. Quarteroni, A. Veneziani and C. Vergara, Geometric multiscale modeling of the cardiovascular system, between theory and practice. Comput. Methods Appl. Mech. Eng. 302 (2016) 193–252. [Google Scholar]
  40. G. Raimondi, Computational models for root water uptake. Master’s thesis, Politecnico di Milano (2017). [Google Scholar]
  41. S.A. Sauter and R. Warnke, Extension operators and approximation on domains containing small geometric details. East-West J. Numer. Math. 7 (1999) 61–77. [Google Scholar]
  42. T. Secomb, R. Hsu, E. Park and M. Dewhirst, Green’s function methods for analysis of oxygen delivery to tissue by microvascular networks. Ann. Biomed. Eng. 32 (2004) 1519–1529. [CrossRef] [PubMed] [Google Scholar]
  43. M. Solomyak, On approximation of functions from Sobolev spaces on metric graphs. J. Approx. Theor. 121 (2003) 199–219. [CrossRef] [Google Scholar]
  44. A.-K. Tornberg and B. Engquist, Numerical approximations of singular source terms in differential equations. J. Comput. Phys. 200 (2004) 462–488. [Google Scholar]

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