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2019 | Book

Three Faces of Fitz: Science, Communication and Leadership Read chapter Read first chapter

Contributions to Partial Differential Equations and Applications

Editors: Prof. B. N. Chetverushkin, Prof. Dr. W. Fitzgibbon, Prof. Dr. Y.A. Kuznetsov, Prof. Dr. P. Neittaanmäki, Prof. J. Periaux, O. Pironneau

Publisher: Springer International Publishing

Book Series : Computational Methods in Applied Sciences

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book treats Modelling of CFD problems, Numerical tools for PDE, and Scientific Computing and Systems of ODE for Epidemiology, topics that are closely related to the scientific activities and interests of Prof. William Fitzgibbon, Prof. Yuri Kuznetsov, and Prof. O. Pironneau, whose outstanding achievements are recognised in this volume.

It contains 20 contributions from leading scientists in applied mathematics dealing with partial differential equations and their applications to engineering, ab-initio chemistry and life sciences. It includes the mathematical and numerical contributions to PDE for applications presented at the ECCOMAS thematic conference "Contributions to PDE for Applications" held at Laboratoire Jacques Louis Lions in Paris, France, August 31- September 1, 2015, and at the Department of Mathematics, University of Houston, Texas, USA, February 26-27, 2016.

This event brought together specialists from universities and research institutions who are developing or applying numerical PDE or ODE methods with an emphasis on industrial and societal applications.

This volume is of interest to researchers and practitioners as well as advanced students or engineers in applied and computational mathematics. All contributions are written at an advanced scientific level with no effort made by the editors to make this volume self-contained.

It is assumed that the reader is a specialist already who knows the basis of this field of research and has the capability of understanding and appreciating the latest developments in this field.

Table of Contents

Frontmatter
Three Faces of Fitz: Science, Communication and Leadership
Abstract
In what follows we provide a brief overview of the life and work of Professor William Fitzgibbon ( University. of Houston)
Jeff Morgan, Jacques Periaux
Career of Prof. Yuri Kuznetsov
Abstract
In what follows we provide a brief overview of the life and work of Professor Yuri Kuznetsov ( University of Houston)
Boris Chetverushkin, William Fitzgibbon, Jacques Periaux
Olivier Pironneau Career Paper
Abstract
In what follows we provide a brief overview of the life and work of Professor Olivier Pironneau Fitzgibbon ( LJLL Sorbonne Université)
William Fitzgibbon, Jacques Periaux
Mean Field Games for Modeling Crowd Motion
Abstract
We present a model for crowd motion based on the recent theory of mean field games. The model takes congestion effects into account. A robust and efficient numerical method is discussed. Numerical simulations are presented for two examples. The second example, in which all the agents share a common source of risk and have incomplete information, is of particular interest, because it cannot be dealt with without modeling rational anticipation.
Yves Achdou, Jean-Michel Lasry
Remarks About Spatially Structured SI Model Systems with Cross Diffusion
Abstract
One of the simplest deterministic mathematical model for the spread of an epidemic disease is the so-called SI system made of two Ordinary Differential Equations. It exhibits simple dynamics: a bifurcation parameter \(\mathscr {T}_0\) yielding persistence of the disease when \(\mathscr {T}_0 > 1\), else extinction occurs. A natural question is whether this gentle dynamic can be disturbed by spatial diffusion. It is straightforward to check it is not feasible for linear/nonlinear diffusions. When cross diffusion is introduced for suitable choices of the parameter data set this persistent state of the ODE model system becomes linearly unstable for the resulting initial and no-flux boundary value problem. On the other hand “natural” weak solutions can be defined for this initial and no-flux boundary value problem and proved to exist provided nonlinear and cross diffusivities satisfy some constraints. These constraints are not fully met for the parameter data set yielding instability. A remaining open question is: to which solutions does this apply? Periodic behaviors are observed for a suitable range of cross diffusivities.
Verónica Anaya, Mostafa Bendahmane, Michel Langlais, Mauricio Sepúlveda
Automatic Clustering in Large Sets of Time Series
Abstract
To study large sets of interacting time series, we combine spectral analysis of graph Laplacians with simulated annealing to automatically generate optimized clustering of time series, by minimization of cost functions characterizing clustering quality. We apply these techniques to evaluation of connectivity between cortex regions, via analysis of cortex activity recordings by sequences of 3-dimensional fMRI images.
Robert Azencott, Viktoria Muravina, Rasoul Hekmati, Wei Zhang, Michael Paldino
Zero Viscosity Boundary Effect Limit and Turbulence
Abstract
This contribution is based on a theorem of Kato which relates for time dependent problems the appearance of turbulence with the anomalous energy dissipation, giving for the Cauchy problem an avatar of a basic idea of the statistical theory of turbulence. Some variant of this theorem are given and then it is shown how this point of view is in full agreement with several issues of fluid mechanic ranging from Prandtl’s problem to Kutta-Jukowsky’s equations.
Claude Bardos
Parabolic Equations with Quadratic Growth in
Abstract
We study here quasi-linear parabolic equations with quadratic growth in \(\mathbb {R}^{n}\). These parabolic equations are at the core of the theory of PDE; see Friedman (Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, 1964) [6], Ladyzhenskaya et al. (Translations of Mathematical Monographs. AMS, 1968) [4] for details. However, for the applications to physics and mechanics, one deals mostly with boundary value problems. The boundary is often taken to be bounded and the solution is bounded. This brings an important simplification. On the other hand, stochastic control theory leads mostly to problems in \(\mathbb {R}^{n}\). Moreover, the functions are unbounded and the Hamiltonian may have quadratic growth. There may be conflicts which prevent solutions to exist. In stochastic control theory, a very important development deals with BSDE (Backward Stochastic Differential Equations). There is a huge interaction with parabolic PDE in \(\mathbb {R}^{n}\). This is why, although we do not deal with BSDE in this paper, we use many ideas from Briand and Hu (Probab Theory Relat Fields 141(3–4):543–567, 2008) [1], Da Lio and Ley (SIAM J Control Optim 45(1):74–106, 2006) [2], Karoui et al. (Backward stochastic differential equations and applications, Princeton BSDE Lecture Notes, 2009) [3], Kobylanski (Ann Probab 28(2):558–602, 2000) [5]. Our presentation provided here is slightly innovative.
Alain Bensoussan, Jens Frehse, Shige Peng, Sheung Chi Phillip Yam
On the Sensitivity to the Filtering Radius in Leray Models of Incompressible Flow
Abstract
One critical aspect of Leray models for the Large Eddy Simulation (LES) of incompressible flows at moderately large Reynolds number (in the range of few thousands) is the selection of the filter radius. This drives the effective regularization of the filtering procedure, and its selection is a trade-off between stability (the larger, the better) and accuracy (the smaller, the better). In this paper, we consider the classical Leray-\(\alpha \) and a recently introduced (by one of the authors) Leray model with a deconvolution-based indicator function, called Leray-\(\alpha \)-NL. We investigate the sensitivity of the solutions to the filter radius by introducing the sensitivity systems, analyzing them at the continuous and discrete levels, and numerically testing them on two benchmark problems.
Luca Bertagna, Annalisa Quaini, Leo G. Rebholz, Alessandro Veneziani
Model Order Reduction for Problems with Large Convection Effects
Abstract
The reduced basis method allows to propose accurate approximations for many parameter dependent partial differential equations, almost in real time, at least if the Kolmogorov n-width of the set of all solutions, under variation of the parameters, is small. The idea is that any solutions may be well approximated by the linear combination of some well chosen solutions that are computed offline once and for all (by another, more expensive, discretization) for some well chosen parameter values. In some cases, however, such as problems with large convection effects, the linear representation is not sufficient and, as a consequence, the set of solutions needs to be transformed/twisted so that the combination of the proper twist and the appropriate linear combination recovers an accurate approximation. This paper presents a simple approach towards this direction, preliminary simulations support this approach.
Nicolas Cagniart, Yvon Maday, Benjamin Stamm
Parametric Optimization of Pulsating Jets in Unsteady Flow by Multiple-Gradient Descent Algorithm (MGDA)
Abstract
Two numerical methodologies are combined to optimize six design characteristics of a system of pulsating jets acting on a laminar boundary layer governed by the compressible Navier-Stokes equations in a time-periodic regime. The flow is simulated by second-order in time and space finite-volumes, and the simulation provides the drag as a function of time. Simultaneously, the sensitivity equations, obtained by differentiating the governing equations w.r.t. the six parameters are also marched in time, and this provides the six-component parametric gradient of drag. When the periodic regime is reached numerically, one thus disposes of an objective-function, drag, to be minimized, and its parametric gradient, at all times of a period. Second, the parametric optimization is conducted as a multi-point problem by the Multiple-Gradient Descent Algorithm (MGDA) which permits to reduce the objective-function at all times simultaneously, and not simply in the sense of a weighted average.
Jean-Antoine Désidéri, Régis Duvigneau
Mixed Formulation of a Linearized Lubrication Fracture Model in a Poro-elastic Medium
Abstract
We analyse and discretize a mixed formulation for a linearized lubrication fracture model in a poro-elastic medium. The displacement of the medium is expressed in primary variables while the flows in the medium and fracture are written in mixed form, with an additional unknown for the pressure in the fracture. The fracture is treated as a non-planar surface or curve according to the dimension, and the lubrication equation for the flow in the fracture is linearized. The resulting equations are discretized by finite elements adapted to primal variables for the displacement and mixed variables for the flow. Stability and a priori error estimates are derived. A fixed-stress algorithm is proposed for decoupling the computation of the displacement and flow and a numerical experiment is included.
Vivette Girault, Mary F. Wheeler, Kundan Kumar, Gurpreet Singh
Two Decades of Wave-Like Equation for the Numerical Simulation of Incompressible Viscous Flow: A Review
Abstract
A wave-like equation based method for the numerical solution of the Navier-Stokes equations modeling incompressible viscous flow was introduced nearly twenty years ago. From its inception to nowadays it has been applied successfully to the numerical solution of two and three dimensional flow problems for incompressible Newtonian and non-Newtonian viscous fluids, in flow regions with fixed or moving boundaries. The main goals of this article are: (i) To recall the foundations of the wave-like equation methodology, and (ii) to review some typical viscous flow problems where it has been applied successfully.
Roland Glowinski, Tsorng-Whay Pan
Arbitrary Lagrangian-Eulerian Finite Element Method Preserving Convex Invariants of Hyperbolic Systems
Abstract
We present a conservative Arbitrary Lagrangian Eulerian method for solving nonlinear hyperbolic systems. The key characteristics of the method is that it preserves all the convex invariants of the hyperbolic system in question. The method is explicit in time, uses continuous finite elements and is first-order accurate in space and high-order in time. The stability of the method is obtained by introducing an artificial viscosity that is unambiguously defined irrespective of the mesh geometry/anisotropy and does not depend on any ad hoc parameter.
Jean-Luc Guermond, Bojan Popov, Laura Saavedra, Yong Yang
Dual-Primal Isogeometric Tearing and Interconnecting Methods
Abstract
This paper generalizes the Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) method, that is well established as parallel solver for large-scale systems of finite element equations, to linear algebraic systems arising from the Isogemetric Analysis of elliptic diffusion problems with heterogeneous diffusion coefficients in two- and three-dimensional multipatch domains with \(C^0\) smoothness across the patch interfaces. We consider different scalings, and derive the expected polylogarithmic bound for the condition number of the preconditioned systems. The numerical results confirm these theoretical bounds, and show incredibly robustness with respect to large jumps in the diffusion coefficient across the interfaces.
Christoph Hofer, Ulrich Langer
C-Interior Penalty Discontinuous Galerkin Approximation of a Sixth-Order Cahn-Hilliard Equation Modeling Microemulsification Processes
Abstract
Microemulsions can be modeled by an initial-boundary value problem for a sixth order Cahn-Hilliard equation. Introducing the chemical potential as a dual variable, a Ciarlet-Raviart type mixed formulation yields a system consisting of a linear second order evolutionary equation and a nonlinear fourth order equation. The spatial discretization is done by a C\(^0\) Interior Penalty Discontinuous Galerkin (C\(^0\)IPDG) approximation with respect to a geometrically conforming simplicial triangulation of the computational domain. The DG trial spaces are constructed by C\(^0\) conforming Lagrangian finite elements of polynomial degree \(p \ge 2\). For the semidiscretized problem we derive quasi-optimal a priori error estimates for the global discretization error in a mesh-dependent C\(^0\)IPDG norm. The semidiscretized problem represents an index 1 Differential Algebraic Equation (DAE) which is further discretized in time by an s-stage Diagonally Implicit Runge-Kutta (DIRK) method of order \( q \ge 2\). Numerical results show the formation of microemulsions in an oil/water system and confirm the theoretically derived convergence rates.
Ronald H. W. Hoppe, Christopher Linsenmann
On Existence “In the Large” of a Solution to Modified Navier-Stokes Equations
Abstract
The problem on existence “in the large” of a solution to the 3D Navier-Stokes equations is open up to now. Nevertheless, for some modifications of the Navier-Stokes equations describing practical problems this problem has been successfully solved. For instance, for the system of Primitive equations describing large-scale ocean dynamics, existence and uniqueness of a strong solution for any time interval and arbitrary initial conditions and viscosity coefficient was proved (Kobelkov J Math Fluid Mech 9(4):588–610, 2007) [1]. O.A. Ladyzhenskaya proposed (Trudy MIAN SSSR 102:85–104, 1967) [2] a modification of the Navier-Stokes equations allowing to prove existence of a solution “in the large”, but this modification was not “physical”. Here we improve the Ladyzhenskaya result modifying not all the three motion equations, but only two of them and only in two (horizontal) variables (not three). Such kind of problems arises in ocean dynamics models. We also consider the case when the viscosity coefficients in vertical and horizontal directions are different. For all these cases existence “in the large” of a solution is proved. Unfortunately, these results cannot be extended to the case of 3D Navier-Stokes equations as well as in the case of Ladyzhenskaya modification.
George Kobelkov
An Algebraic Solver for the Oseen Problem with Application to Hemodynamics
Abstract
The paper studies an iterative method for algebraic problems arising in numerical simulation of blood flows. Here we focus on a numerical solver for the fluid part of otherwise coupled fluid-structure system of equations which models the hemodynamics in vessels. Application of the finite element method and semi-implicit time discretization leads to the discrete Oseen problem at every time step of the simulation. The problem challenges numerical methods by anisotropic geometry, open boundary conditions, small time steps and transient flow regimes. We review known theoretical results and study the performance of recently proposed preconditioners based on two-parameter threshold ILU factorization of non-symmetric saddle point problems. The preconditioner is applied to the linearized Navier–Stokes equations discretized by the stabilized Petrov–Galerkin finite element (FE) method. Careful consideration is given to the dependence of the solver on the stabilization parameters of the FE method. We model the blood flow in the digitally reconstructed right coronary artery under realistic physiological regimes. The paper discusses what is special in such flows for the iterative algebraic solvers, and shows how the two-parameter ILU preconditioner is able to meet these specifics.
Igor N. Konshin, Maxim A. Olshanskii, Yuri V. Vassilevski
Martin’s Problem for Volume-Surface Reaction-Diffusion Systems
Abstract
We consider a question of global existence for two component volume-surface reaction-diffusion systems. The first of the components diffuses in a region, and then reacts on the boundary with the second component, which diffuses on the boundary. We show that if the first component is bounded a priori on any time interval, and the kinetic terms satisfy a generalized balancing condition, then both solutions exist globally. We also pose an open question in the opposite direction, and give some a priori estimates for associated m component systems.
Jeff Morgan, Vandana Sharma
A Posteriori Error Estimates for the Electric Field Integral Equation on Polyhedra
Abstract
We present a residual-based a posteriori error estimate for the Electric Field Integral Equation (EFIE) on a bounded polyhedron \(\varOmega \) with boundary \(\varGamma \). The EFIE is a variational equation formulated in \({\varvec{H}^{-1/2}_{{{\mathrm{div}}}}(\varGamma )}\). We express the estimate in terms of \(L^2\)-computable quantities and derive global lower and upper bounds (up to oscillation terms).
Ricardo H. Nochetto, Benjamin Stamm
On Some Weighted Stokes Problems: Applications on Smagorinsky Models
Abstract
In this paper we study existence and uniqueness of weak solutions for some non-linear weighted Stokes problems using convex analysis. The characterization of these equations is the viscosity, which depends on the strain rate of the velocity field and in some cases is related with a weight being the distance to the boundary of the domain. Such non-linear relations can be seen as a first approach of mixing-length eddy viscosity from turbulent modeling. A well known model is von Karman’s on which the viscosity depends on the square of the distance to the boundary of the domain. Numerical experiments conclude the work and show properties from the theory.
Jacques Rappaz, Jonathan Rochat
Poincaré Type Inequalities for Vector Functions with Zero Mean Normal Traces on the Boundary and Applications to Interpolation Methods
Abstract
We consider inequalities of the Poincaré–Steklov type for subspaces of \(H^1\)-functions defined in a bounded domain \(\varOmega \in \mathbb {R}^d\) with Lipschitz boundary \(\partial \varOmega \). For scalar valued functions, the subspaces are defined by zero mean condition on \(\partial \varOmega \) or on a part of \(\partial \varOmega \) having positive \(d-1\) measure. For vector valued functions, zero mean conditions are applied to normal components on plane faces of \(\partial \varOmega \) (or to averaged normal components on curvilinear faces). We find explicit and simply computable bounds of constants in the respective Poincaré type inequalities for domains typically used in finite element methods (triangles, quadrilaterals, tetrahedrons, prisms, pyramids, and domains composed of them). The second part of the paper discusses applications of the estimates to interpolation of scalar and vector valued functions on macrocells and on meshes with non-overlapping and overlapping cells.
Sergey Repin
Ensemble Interpretation of Quantum Mechanics and the Two-Slit Experiment
Abstract
An evolution equation model is provided for the two-slit experiment of quantum mechanics. The state variable of the equation is the probability density function of particle positions. The equation has a local diffusion term corresponding to stochastic variation of particles, and a nonlocal dispersion term corresponding to oscillation of particles in the transverse direction perpendicular to their forward motion. The model supports the ensemble interpretation of quantum mechanics and gives descriptive agreement with the Schrödinger equation model of the experiment.
Glenn F. Webb
Correction to: Remarks About Spatially Structured SI Model Systems with Cross Diffusion
Verónica Anaya, Mostafa Bendahmane, Michel Langlais, Mauricio Sepúlveda
Metadata
Title
Contributions to Partial Differential Equations and Applications
Editors
Prof. B. N. Chetverushkin
Prof. Dr. W. Fitzgibbon
Prof. Dr. Y.A. Kuznetsov
Prof. Dr. P. Neittaanmäki
Prof. J. Periaux
O. Pironneau
Copyright Year
2019
Electronic ISBN
978-3-319-78325-3
Print ISBN
978-3-319-78324-6
DOI
https://doi.org/10.1007/978-3-319-78325-3

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