Seminário de Matemática Aplicada e Análise Numérica 

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?

There are many real fluids such as water, which the Newtonianconstitutive equation describes extremely well in almost all flowsituations. Small amounts of polymer additive can have a dramaticeffect on the behavior of these liquids. For example, B. A. Tomsstudied flows in straight pipes and discovered that in theturbulent regime small amounts of polymer additive couldsignificantly reduce the pressure drop necessary to attain a givenflow rate. These changes in behavior are attributed to theviscoelastic nature of the polymeric solution and numerousconstitutive equations have been developed to model these fluids.In contrast, in curved pipes polymer additives were foundexperimentally both to alter the relationship between pressure dropand flow rate in the laminar regime and to alter the criticalReynolds number for transition to turbulence. In this talk, we willdiscuss results for steady, fully developed flows of viscoelasticfluids in curved pipes and contrast this behavior with flows ofNewtonian fluids. Following the approach of W. R. Dean and otherauthors, we have used regular perturbation methods to study flowsof viscoelastic fluids in curved pipes. We have obtained explicitsolutions to the perturbation equations at first order for secondorder fluids and a modified Oldroyd-B fluid. In the absence ofinertial effects, flows of Newtonian fluids in curved pipes do notdisplay a secondary flow, rather a uniaxial flow exists whichdiffers only slightly from the straight pipe solution. In contrast,even in the absence of inertial effects, the class of viscoelasticfluids studied display a secondary motion (see, e.g. Thomas 1963,Bowen et al. 1991, Robertson and Muller 1996). Significantly, for acountable number of combinations of material parameters andReynolds numbers, there is a loss of uniqueness of the solution tothe perturbation equations. For other values of material parametersand Reynolds number, a solution does not even exist. There is aregion in parameter space which is free of such singularities. Thislack of existence to the perturbation equations regardless of themagnitude of the curvature ratio, implies a lack of existence of asolution which is a steady, fully developed perturbation of thestraight pipe solution. The implications of this result are underinvestigation.

In this talk, we review some recent results on fractionalevolution equations of type $D^\alpha_t u+ Au = f$, $t\geq 0$,where $\alpha\in (0,2)$, $\alpha\neq 1$.

First, as a typical example, we consider the fractional Burgersequation. Next, we formulate a general result with $A$$m$-accretive. Some comments on maximal regularity follow; finallywe present an unsolved problem concerning fractional nonlinearhyperbolic equations.

Os problemas de optimização não linear aparecem frequentementeassociados ao controlo em simulação, ao projecto em engenharia e àidentificação de parâmetros e ao ajuste de dados em experimentação.A função objectivo e as restrições apresentam, com regularidade,uma estrutura e um escalonamento próprios, associados àdiscretização de equações diferenciais.

Métodos numéricos modernos em optimização não linear, como sãopor exemplo os casos dos métodos SQP e dos métodos de regiões deconfiança, podem e devem ser adaptados para tirar partido dascaracterísticas da aplicação. Outras situações, como adegenerescência das restrições, por um lado, e a dificuldadecomputacional ou experimental em obter o valor das funções e dassuas derivadas, por outro, colocam sérios desafios aooptimizador.