Applied Mathematics and Numerical Analysis Seminar 

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.

Os métodos iterativos mais recomendados hoje em dia para a resolução de grandes sistemas, pela sua rapidez, e convergência em grande número de casos, são os métodos baseados em subespaços de Krylov. Todos eles podem ser enquadrados em três grupos de acordo com as propriedades de minimização que se impõem.

A ideia base deste tipo de métodos é procurar uma solução aproximada para o sistema $A x = b$ num subespaço de Krylov afim $x_0 + K_m(A,r_0) = \left\{v: v=x_0 + \sum_{i=0}^{m-1} c_i A^i r_0\right\}$, sendo $K_m(A, r_0)$ gerado por $(r_0, A r_0 ,\dots, A^{m-1} r_0 )$, onde $r_0$ é o resíduo de uma solução inicial $x_0$ dada.

No caso do método do Gradiente Conjugado (CG), que também se pode relacionar com o método da descida mais rápida, no subespaço $K_m(A,r_0)$ resolve-se um problema de minimização da norma-$A$ do erro $x_m-x^\ast)^T A(x_m-x^\ast))^{1/2}$. Isto é equivalente a impor que o resíduo de $x_m$ seja ortogonal a qualquer vector de $K_m$.

No caso não simétrico, podem considerar-se vários métodos, entre os quais o mais robusto é o método GMRES, Generalized Minimum RESidual. Este resolve, no subespaço de menor dimensão, um problema de minimização da norma-$2$ do resíduo.

To model the transport of substances (back and forth) between surface and ground water, we study the coupling of the Stokes and the Darcy equations. We formulate the problem as a substructuring or interface problem and we solve it by proposing a) an iterative method similar to the Dirichlet-Neumann; b) a preconditioner for elliptic problems with rapidly varying coefficients. We prove the convergence of the iterative method that we introduce with a classical Banach fixed point method argument. Regarding the numerical analysis, we use the P 1 (cross-grid) - P 0 finite elements for the Stokes problem, while in the porous region we use the classical P 1 finite elements (in other words, introducing a suitable method, we do not need to use the mixed formulation). We can use this formulation, based on classical variational principles, since by using a preconditioner based on homogenized (or effective) coefficients, we replace the problem with oscillating coefficients by another one with constant coefficients. Numerical results for some test cases are also provided.

Short-Course on
Numerical Methods for Nonlinear Diferential Equations and Applications to Physics

1. Numerical Solution of Boundary Value Problems For Nonlinear Ordinary Differential Equations on a Finite Interval
1.1 Linearization method.
1.2 Shooting method.
1.3 Difference pass and differential pass methods.

2. Numerical Solution of Boundary Value Problems for Nonlinear Ordinary Differential Equations on an Infinite Interval
2.1 Example related to electrodynamics: a singular problem for a second-order nonlinear ordinary differential equation.
2.2 Example related to hydrodynamics: a singular problem for a third-order nonlinear ordinary differential equation.

3. Numerical Solution of Boundary Value Problems for Systems of Nonlinear Ordinary Differential Equations on a Finite Interval
3.1 Reduction method applied to Cauchy problems.
3.2 Linearization method.
3.3 Conjugate operator method.

4. Numerical Solution of Boundary Value Problems for Systems of Nonlinear Ordinary Differential Equations on an Infinite Interval
4.1 Example related to hydrodynamics: a flow near a rotating disk of an infinite radius.
4.2 Example related to hydrodynamics: a flow near an immovable infinite base due to a fluid rotation far from the wall (without and in the presence of a magnetic field).

Documentos históricos são fluentes em exemplos de tragédias atribuídas a doenças infecciosas. A transmissão de agentes infecciosos em populações de hospedeiros (humanos ou animais) é um processo dinâmico propenso ao desenvolvimento de modelos matemáticos. Estes vão de simples sistemas de equações diferenciais ordinárias (EDOs) que começaram a ser desenvolvidos no início do século 20, a complicados modelos computacionais cuja popularidade vai crescendo com o aperfeiçoamento de técnicas biológicas e capacidade de computadores.

Começarei por introduzir sistemas de EDOs que servem como simples modelos básicos tradicionalmente utilizados por epidemiologistas na interpretação de padrões epidemiológicos, os seus determinantes e possíveis estratégias de contrôlo. Depois demonstrarei a construção de variantes destinadas a classes de doenças com determinadas características. Os resultados serão descritos e comparados com os modelos anteriores.

We consider the numerical solution of some Volterra integro-differential equations of the form: \[ y'(t)= g(t)= \int_0^t k(t,s,y(s)) ds \] By careful choice of the original equation we can give an analysis that shows four distinct types of behaviour in the exact solution. The challenge for the numerical methods is then to show that we obtain the correct behaviour in the numerical solution for each of these types of true behaviour. We focus on simple numerical methods and give diagrams that illustrate how well each method performs.

The Navier-Stokes equations are known since the 19th century. They were derived under the assumption that the fluid is a continuous medium, under Newton's law, and moreover under an a priori assumption that velocity and pressure have a certain smoothness. The existence of solutions with this smoothness in a 3D-case still remains an open mathematical problem!

Comment: There exist mechanisms in real fluids which do not enable the speed of motion to increase above all limits. It is highly desirable to know whether the Navier-Stokes model (a good model?) also involves such mechanisms or whether it admits solutions with singularities?