Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Os escoamentos altamente convectivos representam um desafio na simulação pelo método de elementos finitos. Com a solução de elementos finitos de Galerkin para escoamentos incompressíveis, a matriz associada ao termo convectivo é não simétrica, e portanto, a propiedade de aproximação ótima é perdida. Na prática as soluções apresentam oscilações espúrias. Muitos métodos foram desenvolvidos com o fim de resolver esse problema. Neste trabalho apresentamos um método semi- Lagrangeano, o qual é implicitamente um método do tipo upwind, que portanto resolve o problema anterior, e comparamos o desempenho do método na solução das equações de convecção-difusão e Navier-Stokes incompressível com o Streamline Upwind Petrov Galerkin (SUPG), um método estabilizador de reconhecido desempenho. No SUPG, as funções de forma e de teste são tomadas em espaços diferentes, criando um efeito tal que as oscilações espúrias são drasticamente atenuadas. O método semi-Lagrangeano é um método de fator de integração, no qual o fator é um operador de convecção que se desloca para um sistema de coordenadas móveis no fluido, mas restabelece o sistema de coordenadas Lagrangeanas depois de cada passo de tempo. Isto prevê estabilidade e a possibilidade de utilizar passos de tempo maiores.Existem muitos trabalhos na literatura analisando métodos estabilizadores, mas não assim com o método semi-Lagrangeano, o que representa a contribuição principal deste trabalho: reconhecer as virtudes e as fraquezas do método semi-Lagrangeano em escoamentos dominados pelo fenômeno de convecção. / Convection dominated flows represent a challenge for finite element method simulation. Many methods have been developed to address this problem. In this work we compare the performance of two methods in the solution of the convectiondiffusion and Navier-Stokes equations on environmental flow problems: the Streamline Upwind Petrov Galerkin (SUPG) and the semi-Lagrangian method. In Galerkin finite element methods for fluid flows, the matrix associated with the convective term is non-symmetric, and as a result, the best approximation property is lost. In practice, solutions are often corrupted by espurious oscillations. In this work, we present a semi- Lagrangian method, which is implicitly an upwind method, therefore solving the spurious oscillations problem, and a comparison between this semi-Lagrangian method and the Streamline Upwind Petrov Galerkin (SUPG), an stabilizing method of recognized performance. The SUPG method takes the interpolation and the weighting functions in different spaces, creating an effect so that the spurious oscillations are drastically attenuated. The semi-Lagrangean method is a integration factor method, in which the factor is an operator that shifts to a coordinate system that moves with the fluid, but it resets the Lagrangian coordinate system after each time step. This provides stability and the possibility to take bigger time steps. There are many works in the literature analyzing stabilized methods, but they do not analyze the semi-Lagrangian method, which represents the main contribution of this work: to recognize the strengths and weaknesses of the semi-Lagrangian method in convection dominated flows.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Os escoamentos altamente convectivos representam um desafio na simulação pelo método de elementos finitos. Com a solução de elementos finitos de Galerkin para escoamentos incompressíveis, a matriz associada ao termo convectivo é não simétrica, e portanto, a propiedade de aproximação ótima é perdida. Na prática as soluções apresentam oscilações espúrias. Muitos métodos foram desenvolvidos com o fim de resolver esse problema. Neste trabalho apresentamos um método semi- Lagrangeano, o qual é implicitamente um método do tipo upwind, que portanto resolve o problema anterior, e comparamos o desempenho do método na solução das equações de convecção-difusão e Navier-Stokes incompressível com o Streamline Upwind Petrov Galerkin (SUPG), um método estabilizador de reconhecido desempenho. No SUPG, as funções de forma e de teste são tomadas em espaços diferentes, criando um efeito tal que as oscilações espúrias são drasticamente atenuadas. O método semi-Lagrangeano é um método de fator de integração, no qual o fator é um operador de convecção que se desloca para um sistema de coordenadas móveis no fluido, mas restabelece o sistema de coordenadas Lagrangeanas depois de cada passo de tempo. Isto prevê estabilidade e a possibilidade de utilizar passos de tempo maiores.Existem muitos trabalhos na literatura analisando métodos estabilizadores, mas não assim com o método semi-Lagrangeano, o que representa a contribuição principal deste trabalho: reconhecer as virtudes e as fraquezas do método semi-Lagrangeano em escoamentos dominados pelo fenômeno de convecção. / Convection dominated flows represent a challenge for finite element method simulation. Many methods have been developed to address this problem. In this work we compare the performance of two methods in the solution of the convectiondiffusion and Navier-Stokes equations on environmental flow problems: the Streamline Upwind Petrov Galerkin (SUPG) and the semi-Lagrangian method. In Galerkin finite element methods for fluid flows, the matrix associated with the convective term is non-symmetric, and as a result, the best approximation property is lost. In practice, solutions are often corrupted by espurious oscillations. In this work, we present a semi- Lagrangian method, which is implicitly an upwind method, therefore solving the spurious oscillations problem, and a comparison between this semi-Lagrangian method and the Streamline Upwind Petrov Galerkin (SUPG), an stabilizing method of recognized performance. The SUPG method takes the interpolation and the weighting functions in different spaces, creating an effect so that the spurious oscillations are drastically attenuated. The semi-Lagrangean method is a integration factor method, in which the factor is an operator that shifts to a coordinate system that moves with the fluid, but it resets the Lagrangian coordinate system after each time step. This provides stability and the possibility to take bigger time steps. There are many works in the literature analyzing stabilized methods, but they do not analyze the semi-Lagrangian method, which represents the main contribution of this work: to recognize the strengths and weaknesses of the semi-Lagrangian method in convection dominated flows.

This thesis describes several approaches for stabilizing POD-ROMs (that is, reduced order models based on basis functions derived from the proper orthogonal decomposition) for both the CDR (convection-diffusion-reaction) equation and the NSEs (Navier-Stokes equations). Stabilization is necessary because standard POD-ROMs of convection-dominated problems usually display numerical instabilities. The first stabilized ROM investigated is a streamline-upwind Petrov-Galerkin ROM (SUPG-ROM). I prove error estimates for the SUPG-ROM and derive optimal scalings for the stabilization parameter. I test the SUPG-ROM with the optimal parameter in the numerical simulation of a convection-dominated CDR problem. The SUPG-ROM yields more accurate results than the standard Galerkin ROM (G-ROM) by eliminating the inherent numerical artifacts (noise) in the data and dampening spurious oscillations. I next propose two regularized ROMs (Reg-ROMs) based on ideas from large eddy simulation and turbulence theory: the Leray ROM (L-ROM) and the evolve-then-filter ROM (EF-ROM). Both Reg-ROMs use explicit POD spatial filtering to regularize (smooth) some of the terms in the standard G-ROM. I propose two different POD spatial filters: one based on the POD projection and a novel POD differential filter. These two new Reg-ROMs and the two spatial filters are investigated in the numerical simulation of the three-dimensional flow past a circular cylinder problem at Re = 100. The numerical results show that EF-ROM-DF is the most accurate Reg-ROM and filter combination and the differential filter generally yields better results than the projection filter. The Reg-ROMs perform significantly better than the standard G-ROM and decrease the CPU time (compared against the direct numerical simulation) by orders of magnitude (from about four days to four minutes). / Ph. D.

Optimal control problems (OCPs) governed by convection dominated diffusion-convection-reaction equations arise in many science and engineering applications such as shape optimization of the technological devices, identification of parameters in environmental processes and flow control problems. A characteristic feature of convection dominated optimization problems is the presence of sharp layers. In this case, the Galerkin finite element method performs poorly and leads to oscillatory solutions. Hence, these problems require stabilization techniques to resolve boundary and interior layers accurately. The Streamline Upwind Petrov-Galerkin (SUPG) method is one of the most popular stabilization technique for solving convection dominated OCPs. The focus of this thesis is the application and analysis of the SUPG method for distributed and boundary OCPs governed by evolutionary diffusion-convection-reaction equations. There are two approaches for solving these problems: optimize-then-discretize and discretize-then-optimize. For the optimize-then-discretize method, the time-dependent OCPs is transformed to a biharmonic equation, where space and time are treated equally. The resulting optimality system is solved by the finite element package COMSOL. For the discretize-then-optimize approach, we have used the so called allv at-once method, where the fully discrete optimality system is solved as a saddle point problem at once for all time steps. A priori error bounds are derived for the state, adjoint, and controls by applying linear finite element discretization with SUPG method in space and using backward Euler, Crank- Nicolson and semi-implicit methods in time. The stabilization parameter is chosen for the convection dominated problem so that the error bounds are balanced to obtain L2 error estimates. Numerical examples with and without control constraints for distributed and boundary control problems confirm the effectiveness of both approaches and confirm a priori error estimates for the discretize-then-optimize approach.

Problems governed by partial differential equations (PDEs) in deformable domains, t Rd; d = 2; 3; are of fundamental importance in science and engineering. They are of particular relevance in the design of many engineering systems e.g., aircrafts and bridges as well as to the analysis of several biological phenomena e.g., blood ow in arteries. However, developing numerical scheme for such problems is still very challenging even when the deformation of the boundary of domain is prescribed a priori. Possibility of excessive mesh distortion is one of the major challenge when solving such problems with numerical methods using boundary tted meshes. The arbitrary Lagrangian- Eulerian (ALE) approach is a way to overcome this difficulty. Numerical simulations of convection-dominated problems have for long been the subject to many researchers. Galerkin formulations, which yield the best approximations for differential equations with high diffusivity, tend to induce spurious oscillations in the numerical solution of convection dominated equations. Though such spurious oscillations can be avoided by adaptive meshing, which is computationally very expensive on ne grids. Alternatively, stabilization methods can be used to suppress the spurious oscillations. In this work, the considered equation is designed within the framework of ALE formulation. In the first part, Streamline Upwind Petrov-Galerkin (SUPG) finite element method with conservative ALE formulation is proposed. Further, the first order backward Euler and the second order Crank-Nicolson methods are used for the temporal discretization. It is shown that the stability of the semi-discrete (continuous in time) ALE-SUPG equation is independent of the mesh velocity, whereas the stability of the fully discrete problem is unconditionally stable for implicit Euler method and is only conditionally stable for Crank-Nicolson time discretization. Numerical results are presented to support the stability estimates and to show the influence of the SUPG stabilization parameter in a time-dependent domain. In the second part of this work, SUPG stabilization method with non-conservative ALE formulation is proposed. The implicit Euler, Crank-Nicolson and backward difference methods are used for the temporal discretization. At the discrete level in time, the ALE map influences the stability of the corresponding discrete scheme with different time discretizations, and it leads to schemes where conservative and non-conservative formulations are no longer equivalent. The stability of the fully discrete scheme, irrespective of the temporal discretization, is only conditionally stable. It is observed from numerical results that the Crank-Nicolson scheme induces high oscillations in the numerical solution compare to the implicit Euler and the backward difference time discretiza-tions. Moreover, the backward difference scheme is more sensitive to the stabilization parameter k than the other time discretizations. Further, the difference between the solutions obtained with the conservative and non-conservative ALE forms is significant when the deformation of domain is large, whereas it is negligible in domains with small deformation. Finally, the local projection stabilization (LPS) and the higher order dG time stepping scheme are studied for convection dominated problems. The analysis is based on the quadrature formula for approximating the integrals in time. We considered the exact integration in time, which is impractical to implement and the Radau quadrature in time, which can be used in practice. The stability and error estimates are shown for the mathematical basis of considered numerical scheme with both time integration methods. The numerical analysis reveals that the proposed stabilized scheme with exact integration in time is unconditionally stable, whereas Radau quadrature in time is conditionally stable with time-step restriction depending on the ALE map. The theoretical estimates are illustrated with appropriate numerical examples with distinct features. The second order dG(1) time discretization is unconditionally stable while Crank-Nicolson gives the conditional stable estimates only. The convergence order for dG(1) is two which supports the error estimate.