Applied Mathematics and Numerical Analysis Seminar 

A sedimentary basin is a large porous medium (several hundred kilometers in length and width and five kilometers in depth) which evolves in the course of time with the sedimentation and compaction effects. During this evolution, hydrocarbons appear and flow in this basin. We establish a model which allows for the simulation of a sedimentary basin evolution (compaction, sedimentation) and the hydrocarbon flows generation, migration and trapping. These phenomena occur during millions of years. Consequently, we mainly study time discretization of these equations. In order to solve the corresponding system of strongly coupled equations, we use an explicit scheme in time which is known in the petroleum world as IMPES (Implicit Pressure Explicit Saturations) and the 5 points Finite Volume method for the space discretization. As we concentrate our attention on the time discretization, we use a cartesian grid to mesh the rectangular domain. Due to the explicit scheme used and the nonlinear equations, a C.F.L. condition appears. Therefore, we develop an empirical time strategy which is based on flux throughout the discretization cells. This strategy permits to reduce CPU-time. Nevertheless, we remark that we lose computational time due to local phenomena, so we exhibit a time local refinement and time adaptive strategy. Adequate aposteriori estimators are obtained for the Finite Volume method developed in the basin simulator. Finally, we end this talk by showing a space adaptive mesh strategy which uses the previously developed aposteriori estimators. The main idea of this strategy consists to distinguish the regions where the solution is well computed and those where an improvement of the accuracy is necessary. In the latter, the accuracy is improved by means of a time step refinement and a new co mputation of all the quantities. Nethertheless, the distinction between good and bad areas constitues a very serious difficulty. We overcome this issue using adequate a posteriori estimators for which we obtain several theorical and numerical results in the case of linear parabolic equations. These a posteriori estimators are obtained for the Finite Volume method developped in our basin simulator. Finally, we end this talk by showing a space adaptive mesh strategy which uses the previously developed aposteriori estimators.

This talk is dedicated to high rotating flows in a cylinder. After recalling the state of the art on low Rossby and Froude number problems in a periodic domain or in the whole space, we explain where the Schiffer’s conjecture appears in the bounded cylinder case.

We follow the early historical roots and give late mathematical explanations for some of the lowest-order “dream elements” for thin structures. The tour covers beams, arches, plates and shells.

This work solves the problem governing the simultaneous motion of two viscous liquids of different kinds: compressible and incompressible. The boundary between the fluids is considered as an unknown (free) interface where the surface tension is taken into account. Although the fluids occupy the whole space $\mathbb{R}^3$, one of them should have a finite volume. Local (in time) unique solvability of this problem is obtained in the Sobolev--Slobodetskii spaces of functions. Estimates of the solution of a model problem for the Stokes equations are considered in detail, the interface between the fluids being a plane. The Schauder method is used to study a linear problem with a compact boundary. The passage to the nonlinear problem is made by successive approximations.

In this lecture, we will cover the fundamental concepts behind constitutive modeling of the arterial wall. We will begin with a brief description of the arterial wall structure, focusing on the behavior of elastine and collagene, the components responsible for the passive mechanical strength of the wall. We will then briefly review some fundamentals in continuum mechanics, necessary for discussing nonlinear elastic constitutive equations used for modeling the arterial wall. This will be followed by a discussion of some of the most commonly used nonlinear constitutive models as well as a novel inelastic model introduced in our research group. We will then cover some commonly used experimental methods for measuring material constants found in the nonlinear models. We will end with a discussion of some outstanding problems in the field.

Let $\Omega$ be a three-dimensional exterior domain of class $C^2$. We consider the following Navier-Stokes problem: \[\begin{equation}\begin{gathered}\partial_t v+v\cdot \operatorname{grad}v = \Delta v+\operatorname{grad}p+\operatorname{div}f, \quad \operatorname{div} v = 0, \text{ in } \Omega\times (0,\infty) \\ v(x,t)|_{\partial\Omega}=0, \quad \lim_{|x|\to\infty} v(x,t)=0, \quad t\in [0,\infty),\end{gathered} \label{eq1:522}\end{equation}\] where $f=\{f_{ij}(x,t)\}$ is a second-order tensor field such that $f(x,t)=f(x,t+T)$, for all $t\geq 0$, and some $T\gt 0$.

The objective of this talk is to show that, if $f$ satisfies suitable regularity conditions and its norm, in appropriate function spaces, is sufficiently small, problem ($\ref{eq1:522}$) admits at least one time-periodic strong solution $v, p$. Moreover, the velocity field $v$ decays to zero for large $x$ as $|x|^{-1}$ and its spatial gradient decays as $|x|^{-2}$, both uniformly in time. In addition, the pressure $p$ decays like $|x|^{-2}$ and its gradient like $|x|^{-3}$, for almost all $t\in [0,T]$. If, in particular, $f$ is time-independent, the corresponding solution is also time-independent and coincides with Finn's “physically reasonable” solution.

Finally, we show that the above solutions are unique in a class of weak solutions satisfying the “energy inequality” and with corresponding pressure field satisfying certain summability conditions in $\Omega\times [0,T]$.

Some viscoelastic constitutive models, such as the Oldroyd B model, may be described in two ways: a closed-form differential constitutive equation and a kinetic theory description. There are, however, many other constitutive models of polymeric liquids that allow only the latter form. These (mesoscopic) models are generally regarded as being potentially more realistic than their closed-form counterparts but their numerical simulation may require much more work. Mathematically, these models can be written in two equivalent forms: either as stochastic differential equations or as deterministic Fokker-Planck (FP) equations. The first option gives rise to stochastic numerical methods, which have become very popular during the last 10 years (CONNFFESSIT method, Brownian configuration fields etc.) The second option is relatively unexploited. In this seminar we will consider the application of FP-based methods to the solution of flows of dilute polymeric solutions where the polymers are represented as FENE dumbbells. The seminar is divided into two parts: (i) We begin by describing some FP-based methods with the usual homogeneous flow assumption (over an ensemble of dumbbells) that enables the velocity of a fluid at any point in the flow domain to be written as the linear part of a Taylor series about a reference point (ie the velocity gradient is a constant). We note the considerable saving in CPU time over conventional stochastic techniques that is realisable for low-order configurational space. (ii) We then consider the consequences of a departure from the usual homogeneous flow assumption. This leads to an FP equation for the configurational distribution function (cdf) with diffusion terms in both real space and configurational space. Thus, unlike the case of homogeneous flows, boundary conditions on the cdf must be found. The modified Fokker-Planck equation for the cdf is solved in both physical and configurational space with appropriate boundary conditions and proper account is taken of the fact that configurational space will change as a function of physical position. On this latter point, it has usually been assumed that for dumbbells in two-dimensional flow the configurational space is a disc with radius the maximum extensibility of the dumbbell. However it is clear that within a molecule distance of a physical solid boundary the dumbbell (or, more realistically, the chain) is restricted in the configurations that it may assume. The seminar will conclude with a brief overview of extensions of the above methods to high-order configurational space and to the simulation of flows of melts and concentrated polymeric solutions.

Transparent conditions on open boundaries of a wind tunnel for time-dependent transonic flow problems are proposed. Under assumption that the far field is described by the Euler equations linearized about the free-stream flow, we obtain exact conditions on inflow and outflow cross-sections of a computational domain. The exact conditions are non-local; therefore a special attention is paid to the efficient numerical implementation of our formulae. Examples of numerical calculations are discussed.

In the classification of shells, among the two classical families known as membrane dominated and bending dominated, there is a third class which is surely less known. For such family, theoretical results about the energy norm behavior and percentage of bending energy have been proved rather recently. Such results allow a rigorous study of the asymptotic behavior of some "famous" shell benchmarks (in other words, problems which are classical tests for shell finite elements in the engineering literature). In the first part of the talk I will therefore introduce briefly the classical theory of shells and follow with a survey of the recent aforementioned results. In the second part I will show the theoretical results obtained for the classical shell benchmarks known as "pinched roof" and "Scordelis-Lo roof", and compare them with those obtained through numerical techniques.

First we study the regularity properties of linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. We then use these results in the analysis of piecewise polynomial collocation methods for solving such equations numerically.

Let $\Omega\subset \mathbb{R}^3$ be a domain with smooth boundary $\partial\Omega$. We investigate a mixed boundary value problem for the following strongly elliptic system of second order differential equations \[ \begin{aligned} & S_{\epsilon} u = \left(-\Delta v+ \nabla p, - {\epsilon}^2 \Delta p + \operatorname{div} v \right)=(f',f_4) \quad \text{ in }\Omega, \\ & v= g',\quad \partial_n p=g_4 \quad \text{ on }\partial \Omega, \end{aligned} \qquad (\text{ S$_\epsilon$ }) \] where we focus our interest on asymptotically precise estimates for the solutions describing their behavior as $\epsilon \to 0$. This system ought be considered as a singular perturbation of the Stokes system (S$_0$) which appears if we set $\epsilon=0$ and cancel the Neumann boundary condition for $p$, in particular the type of ellipticity is changed with $\epsilon =0$. With $f_4=0$ and vanishing boundary values, the above system appears in numerical schemes for the Navier-Stokes equations on bounded domains, namely, in the so-called pressure-stabilization methods. If $\Omega$ is a bounded domain, the energy methods applied there to estimate the error between the solution $(v^{\epsilon},p^{\epsilon})$ of the system (S$_{\epsilon}$) and the solution $(v^0,p^0)$ to the Stokes system are of the form \[ \|v^{\epsilon} -v^0; H^1(\Omega)\| + \|p^{\epsilon} -p^0;L^2(\Omega)_\bot\| \leq C\,\epsilon\, \|f';L^2(\Omega)\|. \] (the index $\bot$ stands for functions with vanishing mean value).

To obtain asymptotically precise estimates we introduce Sobolev norms depending on the small parameter $\epsilon \gt 0$. It turns out that for bounded domains, under the additional smoothness assumption $f' \in H^1(\Omega)$, these estimates can be improved up to convergence of $v^{\epsilon}$ in $H^2(\Omega)$ and $p^\varepsilon$ in $H^1(\Omega)_\bot$.

Apparently up to now the corresponding nonlinear problems as well as the case of unbounded domains were not yet considered. Here the interest is focused on the exterior Dirichlet problem for the linear systems. The appropriate function spaces for the investigations are step weighted, parameter dependent Sobolev spaces. This leads to asymptotically precise estimates as $\epsilon \to 0^{+}$, and enables us to derive the complete asymptotics of the solutions to (S$_\epsilon$) as $|x| \to \infty$. The latter is remarkable insofar as this system itself for $\epsilon \gt 0$ the usual arguments of Kondratiev theory, the differential operator $S_\epsilon$ is not admissible at infinity.

The results presented here are obtained in a joint work with S. A. Nazarov, St. Petersburg.

Problemas Diretos de transferência de calor e massa envolvem a determinação dos campos de temperatura e/ou concentração, a partir do conhecimento da geometria do problema, das condições de contorno e inicial, dos termos-fonte presentes na região e das propriedades físicas envolvidas na formulação. Problemas diretos, formulados de maneira apropriada, são matematicamente classificados como bem-postos, isto é, a solução do problema satisfaz os critérios de existência, unicidade e estabilidade com relação aos dados de entrada.

Por outro lado, Problemas Inversos envolvem a estimativa de pelo menos um dos parâmetros/funções que aparecem na formulação do problema direto bem-posto, a partir do conhecimento dos valores de temperatura e/ou concentração em pontos apropriados na região em estudo. Por exemplo, o fluxo de calor na superfície de um satélite em reentrada na atmosfera pode ser estimado a partir de medidas de temperatura tomadas no interior do material de proteção térmica. Assim, evita-se a localização de sensores na superfície do satélite, os quais podem ser danificados ou fornecer medidas imprecisas durante o procedimento de reentrada atmosférica.

Problemas inversos são de um modo geral matematicamente classificados como mal-postos. A solução de um problema inverso normalmente não satisfaz o critério de estabilidade, onde pequenas perturbações nos dados de entrada (erros nas medidas experimentais de temperatura e/ou concentração, por exemplo) podem ser amplificadas. Portanto, para a obtenção da solução de um problema inverso é necessário reformulá-lo em termos de um problema bem-posto, utilizando-se técnicas apropriadas de regularização (suavização). Problemas inversos podem ser resolvidos através de técnicas de minimização para a estimativa de parâmetros, onde a minimização é realizada em um espaço de dimensão finita, ou para a estimativa de funções, onde a minimização é realizada em um espaço de funções de dimensão infinita.

Neste seminário serão abordados métodos de solução de problemas inversos de estimativa de parâmetros e de estimativa de funções, incluindo o Método de Levenberg-Marquardt e o Método do Gradiente Conjugado. Serão também apresentados resultados obtidos na solução de problemas inversos de interesse prático, incluindo a estimativa das componentes de condutividade térmica de sólidos ortotrópicos; dos parâmetros adimensionais da formulação de Luikov para transferência de calor e massa em meios porosos capilares; bem como do fluxo de calor nas paredes de canais ou cavidades, em problemas de convecção forçada e natural, respectivamente.