A Priori Estimations of a Global Homotopy Residue Continuation Method

International audienceThis work is concerned with the a priori estimations of a global homotopy residue continuation method starting from a disjoint initial guess. Explicit conditions ensuring the quadratic convergence of the underlying Newton-Raphson algorithm are proved

Continuation methods and disjoint equilibria

International audienceContinuation methods are efficient to trace branches of fixed point solutions in parameter space as long as these branches are connected. However, the computation of isolated branches of fixed points is a crucial issue and require ad-hoc techniques. We suggest a modification of the standard continuation methods to determine these isolated branches more systematically. The so-called residue continuation method is a global homotopy starting from an arbitrary disjoint initial guess. Explicit conditions ensuring the quadratic convergence of the underlying Newton-Raphson process are derived and illustrated through several examples

Collocation methods for complex delay models of structured populations

openDottorato di ricerca in Informatica e scienze matematiche e fisicheopenAndo', Alessi

An interface formulation of the Laplace-Beltrami problem on piecewise smooth surfaces

The Laplace-Beltrami problem on closed surfaces embedded in three dimensions arises in many areas of physics, including molecular dynamics (surface diffusion), electromagnetics (harmonic vector fields), and fluid dynamics (vesicle deformation). In particular, the Hodge decomposition of vector fields tangent to a surface can be computed by solving a sequence of Laplace-Beltrami problems. Such decompositions are very important in magnetostatic calculations and in various plasma and fluid flow problems. In this work we develop $L^2$-invertibility theory for the Laplace-Beltrami operator on piecewise smooth surfaces, extending earlier weak formulations and integral equation approaches on smooth surfaces. Furthermore, we reformulate the weak form of the problem as an interface problem with continuity conditions across edges of adjacent piecewise smooth panels of the surface. We then provide high-order numerical examples along surfaces of revolution to support our analysis, and discuss numerical extensions to general surfaces embedded in three dimensions

Deflation techniques for finding distinct solutions of nonlinear partial differential equations

Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find starting points that lie in different basins of attraction. In this paper, we present an infinite-dimensional deflation algorithm for systematically modifying the residual of a nonlinear PDE problem to eliminate known solutions from consideration. This enables the Newton-Kantorovitch iteration to converge to several different solutions, even starting from the same initial guess. The deflated Jacobian is dense, but an efficient preconditioning strategy is devised, and the number of Krylov iterations are observed not to grow as solutions are deflated. The power of the approach is demonstrated on several problems from special functions, phase separation, differential geometry and \ud fluid mechanics that permit distinct solutions

Benchmarking the Immersed Boundary Method for Viscoelastic Flows

We present and analyze a series of benchmark tests regarding the application of the immersed boundary (IB) method to viscoelastic flows through and around non-trivial, stationary geometries. The IB method is widely used for the simulation of biological fluid dynamics and other modeling scenarios where a structure is immersed in a fluid. Although the IB method has been most commonly used to model systems with viscous incompressible fluids, it also can be applied to visoelastic fluids, and has enabled the study of a wide variety of dynamical problems including the settling of vesicles and the swimming of elastic filaments in fluids modeled by the Oldroyd-B constuitive equation. However, to date, relatively little work has explored the accuracy or convergence properties of the numerical scheme. Herein, we present benchmarking results for an IB solver applied to viscoelastic flows in and around non-trivial geometries using the idealized Oldroyd-B and more realistic, polymer-entanglement-based Rolie-Poly constitutive equations. We use two-dimensional numerical test cases along with results from rheology experiments to benchmark the IB method and compare it to more complex finite element and finite volume viscoelastic flow solvers. Additionally, we analyze different choices of regularized delta function and relative Lagrangian grid spacings which allow us to identify and recommend the key choices of these numerical parameters depending on the present flow regime.Comment: 31 pages, 15 figure

Elliptic operators, connections and gauge transformations

A study is made of the action of various Banach Lie groups of principal bundle automorphims (gauge transformations) on corresponding spaces of connections on some principal bundle, using standard theorems of global analysis together with elliptic regularity theorems. A proof of elliptic regularity theorems in Sobolev and Holder norms for linear elliptic partial differential operators with smooth coefficients acting on sections of smooth vector bundles is presented. This proofassumes acquaintance with the theory of tempered distributions and their Fourier transforms and with the theory of compact and Fredholm operators, and also uses results from the papers of Calderon and Zygmund and from the early papers of Hormander on pseuoo-differential operators, but is otherwise intended to be self-contained. Elliptic regularity theorems arc proved for elliptic orcrators with non-smooth coefficients, using only the regularity theorems for elliptic operators with smooth coefficients, together with the Sobolev embedding theorems, the Rellich-Kondrakov theorem and the Sobolev multiplication theorems.For later convenience these elliptic regularity results are presented as a generalization of the analytical aspects of Hodge theory. Various theorems concerning the action of automorphisms on connections are proved, culminating in the slice theorems obtained in chapter VIII. Regularity theorems for Yang-Mills connections and for Yang-Mills Higgs systems arc obtained, In chapter IX analytical properties of the covariant derivative operators associated with a connection arc related to the holonomy group of the connection via a theorem which shows the existence of an upper bound on the length of loop required Lo generate the holonomy group of a connection with compact holonomygroup