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

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.

[1]  D. W. Peaceman.  Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks and anisotropic permeability , 1983 .

[2]  P Zunino,et al.  Modelling mass and heat transfer in nano-based cancer hyperthermia , 2015, Royal Society Open Science.

[3]  Carlo D'Angelo,et al.  Finite Element Approximation of Elliptic Problems with Dirac Measure Terms in Weighted Spaces: Applications to One- and Three-dimensional Coupled Problems , 2012, SIAM J. Numer. Anal..

[4]  Julián Fernández Bonder,et al.  ASYMPTOTIC BEHAVIOR OF THE BEST SOBOLEV TRACE CONSTANT IN EXPANDING AND CONTRACTING DOMAINS , 2002 .

[5]  Barbara I. Wohlmuth,et al.  Optimal A Priori Error Estimates for an Elliptic Problem with Dirac Right-Hand Side , 2014, SIAM J. Numer. Anal..

[6]  M. Arioli,et al.  A finite element method for quantum graphs , 2018 .

[7]  Mikael Mortensen,et al.  Preconditioning trace coupled 3d‐1d systems using fractional Laplacian , 2016, Numerical Methods for Partial Differential Equations.

[8]  P. Zunino,et al.  A computational model of drug delivery through microcirculation to compare different tumor treatments , 2014, International journal for numerical methods in biomedical engineering.

[9]  Kent-André Mardal,et al.  Preconditioners for Saddle Point Systems with Trace Constraints Coupling 2D and 1D Domains , 2016, SIAM J. Sci. Comput..

[10]  Wei Gong,et al.  Approximations of Elliptic Optimal Control Problems with Controls Acting on a Lower Dimensional Manifold , 2014, SIAM J. Control. Optim..

[11]  Laura Cattaneo,et al.  Computational models for fluid exchange between microcirculation and tissue interstitium , 2014, Networks Heterog. Media.

[12]  P. Kuchment,et al.  Quantum Graphs and Their Applications , 2006 .

[13]  B. Engquist,et al.  Numerical approximations of singular source terms in differential equations , 2004 .

[14]  Paolo Zunino,et al.  A computational study of cancer hyperthermia based on vascular magnetic nanoconstructs , 2016, Royal Society Open Science.

[15]  Susanne C. Brenner,et al.  Poincaré-Friedrichs Inequalities for Piecewise H1 Functions , 2003, SIAM J. Numer. Anal..

[16]  S. Sauter,et al.  Extension operators and approximation on domains containing small geometric details , 1999 .

[17]  T. R. Blake,et al.  Analysis of coupled intra- and extraluminal flows for single and multiple capillaries , 1982 .

[18]  H. Weinberger,et al.  An optimal Poincaré inequality for convex domains , 1960 .

[19]  Giustina Casagrande,et al.  A computational model for microcirculation including Fahraeus‐Lindqvist effect, plasma skimming and fluid exchange with the tissue interstitium , 2018, International journal for numerical methods in biomedical engineering.

[20]  Willem F. Bronsvoort,et al.  Self-intersection avoidance and integral properties of generalized cylinders , 2002, Comput. Aided Geom. Des..

[21]  Alexandre Ern,et al.  A Posteriori Control of Modeling Errors and Discretization Errors , 2003, Multiscale Model. Simul..

[22]  J. Nordbotten,et al.  Functional analysis and exterior calculus on mixed-dimensional geometries , 2017, Annali di Matematica Pura ed Applicata (1923 -).

[23]  C. Dangelo,et al.  Multiscale modelling of metabolism and transport phenomena in living tissues , 2007 .

[24]  Barbara I. Wohlmuth,et al.  A Local Error Estimate for the Poisson Equation with a Line Source Term , 2015, ENUMATH.

[25]  Giustina Casagrande,et al.  Numerical simulations of the microvascular fluid balance with a non-linear model of the lymphatic system. , 2019, Microvascular research.

[26]  Rainer Helmig,et al.  A New Simulation Framework for Soil–Root Interaction, Evaporation, Root Growth, and Solute Transport , 2018, Vadose Zone Journal.

[27]  Timothy W. Secomb,et al.  Effect of extravascular pressure gradients on capillary fluid exchange , 1986 .

[28]  Timothy W. Secomb,et al.  Green's Function Methods for Analysis of Oxygen Delivery to Tissue by Microvascular Networks , 2004, Annals of Biomedical Engineering.

[29]  Paolo Zunino,et al.  Mathematical analysis, finite element approximation and numerical solvers for the interaction of 3D reservoirs with 1D wells , 2019, GEM - International Journal on Geomathematics.

[30]  Barbara Wohlmuth,et al.  Mathematical modelling, analysis and numerical approximation of second order elliptic problems with inclusions , 2018 .

[31]  D. W. Peaceman.  Interpretation of Well-Block Pressures in Numerical Reservoir Simulation(includes associated paper 6988 ) , 1978 .

[32]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[33]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[34]  Alfio Quarteroni,et al.  A multiscale Darcy–Brinkman model for fluid flow in fractured porous media , 2011, Numerische Mathematik.

[35]  Timothy W. Secomb,et al.  The interaction of extravascular pressure fields and fluid exchange in capillary networks , 1986 .

[36]  Anna Scotti,et al.  A mixed finite element method for modeling the fluid exchange between microcirculation and tissue interstitium , 2018 .

[37]  Ingeborg G. Gjerde,et al.  Splitting method for elliptic equations with line sources , 2018, ESAIM: Mathematical Modelling and Numerical Analysis.

[38]  Michael Solomyak,et al.  On approximation of functions from Sobolev spaces on metric graphs , 2002, J. Approx. Theory.

[39]  A. Quarteroni,et al.  On the coupling of 1D and 3D diffusion-reaction equations. Applications to tissue perfusion problems , 2008 .

[40]  Silvia Bertoluzza,et al.  Local error estimates of the finite element method for an elliptic problem with a Dirac source term , 2015, 1505.03032.

[41]  Alfio Quarteroni,et al.  Geometric multiscale modeling of the cardiovascular system, between theory and practice , 2016 .