Applied Mathematics and Numerical Analysis Seminar 

In this talk we present and analyse a numerical method for the solution of a distributed order differential equation of the general form $$ \int_0^m \mathcal{A}(r, D^r_*u(t)) dr = f(t) $$ where the derivative $D^r_*$ is taken to be a fractional derivative of Caputo type of order $r$. We give a convergence theory for our method and conclude with some numerical examples.

As is well-known, the Navier-Stokes equations are at the foundations of many branches of applied sciences, including Meteorology, Oceanography, Oil Industry, Airplane, Ship and Car Industries, etc. In each of the above areas, these equations have collected many undisputed successes, which definitely place them among the most accurate, simple and beautiful models of mathematical physics. However, in spite of these successes, to date, a number of unresolved basic questions — mostly, for the physically relevant case of three-dimensional (3D) motions — remain open. Among them, certainly, the most famous is that of proving or disproving existence of 3D regular solutions for all times and for data of arbitrary ‘size’, no matter how smooth. This notorious question has challenged several generations of mathematicians since the beginning of the 20th century who, yet, have not been able two furnish a complete answer. The problem has become so obsessing and intriguing that, as is known, mathematicians have decided to put a generous bounty on it. In fact, properly formulated, it is listed as the third of the seven $1M Millennium Prize Problems of the Clay Mathematical Institute. It should be observed that the analogous question in the two-dimensional (2D) case received a positive answer about half a century ago. In this talk I shall present the main known results of existence, uniqueness and regularity of solutions to the corresponding initial-boundary value problem in a way that should be accessible also to non-specialists. Moreover, I will furnish a number of significant open questions and explain why the current mathematical approaches fail to answer them. In some cases, I shall also point out possible strategies of resolution.

The problem of shape reconstruction of an unknown characteristic source inside a domain is analyzed. We consider a conductivity problem where the heat source is defined as a characteristic function. Restrictions to star-shaped sets arise from a uniqueness theorem by Novikov and are discussed in the context of a Fourier problem. A numerical method based on the reciprocity gap functional for harmonic polynomials and series truncation is proposed to recover the unknown shape from Cauchy noisy data. Extensions to other problems will be discussed. (Joint work with C J S Alves)

This work is concerned with the development of a numerical method capable of simulating viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien-Tanner). In particular, we are interested in flows possessing moving free surfaces. The fluid is modelled by a Marker-and-Cell type method and employs an accurate representation of the fluid surface. Boundary conditions are described in detail and the full free surface stress conditions are considered. The PTT equation is solved by a high order method which requires the calculation of the extra-stress tensor on the mesh contour. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. The numerical method was incorporated into the codes Freeflow2D and Freeflow3D, extending these codes to viscoelastic flows described by the non-linear constitutive equation PTT. To validate the numerical method fully developed flow in a two-dimensional channel was simulated and the numerical solutions were compared with known analytic solutions. The 3D-case was validated by simulating fully developed flow in a 3D-pipe. Convergence results were obtained throughout by using mesh refinement. To demonstrate that complex free surface flows using the PTT model can be computed, extrudate swell and a jet flowing onto a rigid plate were simulated. A short video will be shown. This is joint work with Gilcilene Paulo.

The numerical solution of linear weakly singular Volterra integro-differential equations is discussed. Using an integral equation reformulation of the initial-value problem, we apply to it a smoothing transformation so that the exact solution of the resulting equation does not contain any singularities in its derivatives up to a certain order. After that the regularized equation is solved by a piecewise polynomial collocation method on a mildly graded or uniform grid.

The aim of this work is to extend to compressible and heat conducting flows the well known concept of weak solutions to the incompressible Navier-Stokes equations due to J. Leray in 1933 and extended by P.-L. Lions in 1995 and E. Feireisl in 2001 to barotropic flows. Global existence and stability properties are obained in dimension 2 and 3 for general equation of state including polytropic gas law except close to vacuum. The main idea is to use a new mathematical entropy expressed as some additional Lyapunov function on the whole system which arises when the viscosity coefficients depend in a suitable way on the density. We will compare the results to the one by E. Feireisl where an inequality is obtained on the temperature but viscosity coefficients may depend on temperature. We will also explain why our result help to provide existence of global weak solutions for viscous shallow water systems. This is a joint work with Benoit Desjardins.

An asymptotic analysis of solutions to elliptic problems in domains with rapidly oscillating boundaries (rough and damaged surfaces) will be presented. The construction of several asymptotic terms, including boundary layers, permit for modelling the problem by a much simpler problem, solution of which, nevertheless, gives the two-term asymptotics of the original solution. Two approaches for this type of modelling will be discussed. For the sake of simplicity, boundary value problems for a scalar equation will be considered, although the investigation is directed to the elasticity theory.

Three-dimensional ($3$D) simulations of blood flow provide detailed information important for the comprehension of the cardiovascular system. These consist in coupling the $3$D Navier-Stokes equations with a model for the vessel wall structure. Due to the computational cost of fully $3$D fluid-structure interaction (FSI) problems and to the complexity of the cardiovascular system, these models can be applied only to truncated regions of interest. Following the geometrical multiscale modelling of the cardiovascular system, the remaining parts can be accounting resorting to reduced models, one-dimensional ($1$D) or zero-dimensional ($0$D). In particular, the $1$D models describe very well the wave propagation nature of blood flow and coupled with the $3$D model act as proper absorbing boundary conditions. We address the coupling of $3$D and $1$D FSI models. The difficulty of this coupling lays in putting together such different models, namely it is not evident which conditions to impose at the coupling interface. We study the stability of such coupling, bringing forth proper matching conditions between the models. Different coupling strategies are discussed and several numerical results presented.

Inverse problems for the one-speed equation, transport theory and tomography, simultaneous reconstruction of the absortion and scattering coefficients, source detector methodology.

Variational formulation, transformation of the problem to the self-adjoint form, methods treatement of the angular and spatial dependence, the concept of natural basis for the transport problem, domain partition consistent with the propagation of radiation.

Functional formulation, related Hilbert and Sobolev's spaces, operators, trace, albedo, compactness with the scattering kernel, related eigenvalue problems.

Linear Boltzmann equation, neutral particles transport, some physical concepts, scattering, absortion and fission, particles position, direction of propagation and energy, boundaries in the phase space, boundary values problems.

The solution of realistic problems in Hemodynamics generally involves the solution of very large systems of linear and non-linear equations. These systems arise from the discretization of non-linear partial differential equations, coupling the conservation of mass and momentum with some constitutive equation expressing the relation between stresses and the history of deformation. Analytical solutions are usually not available, either because of the complex geometries of blood vessels or because of the complexity of the constitutive equations. Even considering a simple geometry like a straight tube, exact solutions are only known in the Newtonian case or for very simple shear-dependent fluids like the power law fluids. We present a general method for obtaining solutions for the flow of generalized Newtonian fluids in straigth tubes (2D and 3D), with arbitrary precision, and use these solutions to create benchmarks for numerical solvers. Some numerical examples are shown for the Carreau-Yasuda viscosity model, analyzing the performance of two numerical solvers (Adina and FreeFem++), using different finite element spaces.

Leukocyte recruitment towards and rolling at inflammatory regions on activated endothelial cells has been a subject of intensive experimental and computational investigations. The mechanism of rolling is commonly attributed to active formation and breakage of bonds between selectin adhesion molecules and their ligands. The induced cells first rotate with an angular velocity of about 25 sec-1, before they migrate towards the nearest wall where they roll, deform, adhere and migrate through the endothelial cells. The rolling process may take about 20 seconds during which leukocytes are exposed to chemoattractants. The rolling velocity can be as slow as 2.73 mm/s and as large as 60 mm/s in a typical venule in the rat muscle of 23.5 mm in diameter. Aiming at a better understanding of the complexity of the rolling process and of the influence of hydrodynamic forces, we present here updated experimental and simulation results conducted in collaboration with A. S. Silva and C. Saldanha of FML.

We consider a second order elliptic problem posed on an interval or on a connected polygonal domain in a two dimensional space. We introduce an admissible mesh in the sense of [1]. The convergence order of the finite volume approximate solution (called Basic Finite Volume Solution) is in general $O(h)$ in both discrete norms $L^2$ and $H^1$. We suggest a technique, based on the so-called Fox's Difference Correction [2], which allows us to obtain a new Finite Volume Approximation of order $O(h^{\alpha })$, where $\alpha$ equal to $2$ or $\frac{3}{2}$. In addition, this new Finite Volume Approximation can be computed using the same matrix used to compute the basic finite volume solution. The computational cost is comparable to that of the new Finite Volume Approximation. If the domain problem is an interval or a rectangle, we obtain finite volume approximations of arbitrary order and these approximations can be computed using the same matrix used to compute the Basic Finite Volume Solution. We give an application of our approach to improve the convergence order of finite element solutions defined in non uniform meshes.

References

1. R. Eymard, T. Gallouet and R. Herbin: Finite Volume Methods. Handbook of Numerical Analysis. P. G. Ciarlet and J. L. Lions (eds.), vol. VII, 723-1020, 2000.
2. L. Fox: Some Improvements in the Use of Relaxation Methods for the Solution of Ordinary and Partial Defferential Equations. Proc. Roy. Soc. Lon Ser. A, 190, 31-59, 1947.