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No presente trabalho, utilizou-se o Método da Fronteira Imersa, o qual utiliza dois tipos de malhas computacionais: euleriana (utilizada para o fluido) e lagrangiana (utilizada para representar a interface de separação de dois fluidos). O software livre GMSH foi utilizado para representar um sólido por meio da sua superfície externa e também para gerar uma malha triangular, bidimensional e não estruturada para discretizar essa superfície. Essa superfície foi utilizada como condição inicial para a malha lagrangiana (fronteira imersa). Os dados da malha lagrangiana são armazenados em uma estrutura de dados chamada Halfedge, a qual é largamente utilizada em Computação Gráfica para armazenar superfícies fechadas e orientáveis. Uma vez que a malha lagrangiana esteja armazenada nesta estrutura de dados, passa-se a estudar uma hipotética interação dinâmica entre a fronteira imersa e o escoamento do fluido. Esta interação é estudada apenas em um sentido, considera-se apenas a condição de não deslizamento, isto é, a fronteira imersa acompanhará passivamente um campo de velocidades pré-estabelecido (imposto), sem exercer qualquer força ou influência sobre ele. Foi utilizado um campo de distância local com sinal (função indicadora de fluidos) para identificar o interior e o exterior da superfície que representa a interface entre os fluidos. Este campo de distância é atualizado a cada passo no tempo utilizando idéias de Geometria Computacional, o que tornou o custo computacional para calcular esse campo otimal independente da complexidade geométrica da interface. Esta metodologia mostrou-se robusta e produz uma definição nítida das distintas fases dos fluidos em todos os passos no tempo. Para acompanhar e visualizar de forma mais precisa o comportamento dos fluidos na vizinhança da superfície que representa a interface de separação dos fluido, foi utilizado um algoritmo chamado de Refinamento Adaptativo de Malhas para fazer um refinamento dinâmico da malha euleriana na vizinhança da malha lagrangiana. / The scientific motivation of the present work is the mathematical modeling and the computational simulation of multiphase flows. Specifically, the equations of a two-phase flow are written by combining the Immersed Boundary Method with a suitable fluid indicator function. It is assumed that the fluid equations are discretized on an Eulerian mesh covering completely the flow domain and that the interface between the fluid phases is discretized by a non-structured Lagrangian mesh formed by triangles. In this context, employing tools commonly found in Computational Geometry, the computation of the fluid indicator function is efficiently performed on a block-structured Eulerian mesh bearing dynamical refinement patches. Formed by a set of triangles, the Lagrangian mesh, which is initally generated employing the free software GMSH, is stored in a Halfedge data structure, a data structure which is widely used in Computer Graphics to represent bounded, orientable closed surfaces. Once the Lagrangian mesh has been generated, next, one deals with the hipothetical situation of dealing with the one-way dynamical interaction between the immersed boundary and the fluid flow, that is, considering the non-slip condition, only the action of the flow on the interface is studied. No forces arising on the interface affects the flow, the interface passively being advect with the flow under a prescribed, imposed velocity field. In particular, the Navier-Stokes equations are not solved. The fluid indicator function is given by a signed distance function in a vicinity of the immersed boundary. It is employed to identify interior/exterior points with respect to the bounded, closed region which is assumed to contain one of the fluid phases in its interior. The signed distance is update every time step employing Computational Geometry methods with optimal cost. Several examples in three dimensions, showing the efficiency and efficacy in the computation of the fluid indicator function, are given which employ the dynamical adaptive properties of the Eurlerian mesh for a moving interface.

No presente trabalho, utilizou-se o Método da Fronteira Imersa, o qual utiliza dois tipos de malhas computacionais: euleriana (utilizada para o fluido) e lagrangiana (utilizada para representar a interface de separação de dois fluidos). O software livre GMSH foi utilizado para representar um sólido por meio da sua superfície externa e também para gerar uma malha triangular, bidimensional e não estruturada para discretizar essa superfície. Essa superfície foi utilizada como condição inicial para a malha lagrangiana (fronteira imersa). Os dados da malha lagrangiana são armazenados em uma estrutura de dados chamada Halfedge, a qual é largamente utilizada em Computação Gráfica para armazenar superfícies fechadas e orientáveis. Uma vez que a malha lagrangiana esteja armazenada nesta estrutura de dados, passa-se a estudar uma hipotética interação dinâmica entre a fronteira imersa e o escoamento do fluido. Esta interação é estudada apenas em um sentido, considera-se apenas a condição de não deslizamento, isto é, a fronteira imersa acompanhará passivamente um campo de velocidades pré-estabelecido (imposto), sem exercer qualquer força ou influência sobre ele. Foi utilizado um campo de distância local com sinal (função indicadora de fluidos) para identificar o interior e o exterior da superfície que representa a interface entre os fluidos. Este campo de distância é atualizado a cada passo no tempo utilizando idéias de Geometria Computacional, o que tornou o custo computacional para calcular esse campo otimal independente da complexidade geométrica da interface. Esta metodologia mostrou-se robusta e produz uma definição nítida das distintas fases dos fluidos em todos os passos no tempo. Para acompanhar e visualizar de forma mais precisa o comportamento dos fluidos na vizinhança da superfície que representa a interface de separação dos fluido, foi utilizado um algoritmo chamado de Refinamento Adaptativo de Malhas para fazer um refinamento dinâmico da malha euleriana na vizinhança da malha lagrangiana. / The scientific motivation of the present work is the mathematical modeling and the computational simulation of multiphase flows. Specifically, the equations of a two-phase flow are written by combining the Immersed Boundary Method with a suitable fluid indicator function. It is assumed that the fluid equations are discretized on an Eulerian mesh covering completely the flow domain and that the interface between the fluid phases is discretized by a non-structured Lagrangian mesh formed by triangles. In this context, employing tools commonly found in Computational Geometry, the computation of the fluid indicator function is efficiently performed on a block-structured Eulerian mesh bearing dynamical refinement patches. Formed by a set of triangles, the Lagrangian mesh, which is initally generated employing the free software GMSH, is stored in a Halfedge data structure, a data structure which is widely used in Computer Graphics to represent bounded, orientable closed surfaces. Once the Lagrangian mesh has been generated, next, one deals with the hipothetical situation of dealing with the one-way dynamical interaction between the immersed boundary and the fluid flow, that is, considering the non-slip condition, only the action of the flow on the interface is studied. No forces arising on the interface affects the flow, the interface passively being advect with the flow under a prescribed, imposed velocity field. In particular, the Navier-Stokes equations are not solved. The fluid indicator function is given by a signed distance function in a vicinity of the immersed boundary. It is employed to identify interior/exterior points with respect to the bounded, closed region which is assumed to contain one of the fluid phases in its interior. The signed distance is update every time step employing Computational Geometry methods with optimal cost. Several examples in three dimensions, showing the efficiency and efficacy in the computation of the fluid indicator function, are given which employ the dynamical adaptive properties of the Eurlerian mesh for a moving interface.

Na obtenção de aproximações numéricas para Equações Diferenciais Parciais Elípticas utilizando o Método dos Elementos Finitos (MEF) alguns problemas apresentam valores maiores para o erro somente em algumas determinadas regiões do domínio como, por exemplo, regiões onde existam singularidades na solução contínua do problema. Uma possível alternativa para reduzir o erro cometido nestas regiões é aumentar o número de elementos nos trechos onde o erro cometido foi considerado grande. A questão principal é como identificar essas regiões, dado que a solução do problema contínuo é desconhecida. Neste trabalho iremos apresentar a chamada estimativa residual, que fornece um estimador do erro cometido na aproximação utilizando apenas os valores conhecidos dos contornos e a aproximação obtida sobre uma dada partição de elementos. Vamos discutir a relação entre a estimativa residual e o erro cometido na aproximação, além de utilizar as estimativas na construção de um algoritmo adaptativo para as malhas em estudo. Utilizando o software FreeFem++ serão obtidas aproximações para a Equação de Poisson e para o sistema de equações associado à Elasticidade Linear e por meio do estimador residual será analisado o erro cometido nas aproximações e a necessidade do refinamento adaptativo das malhas. / In obtaining numerical approximations for solutions to Elliptic Partial Differential Equations using the Finite Element Method (FEM) one sees that some problems have higher values for the error only in certain domain regions such as, for example, regions where the solution of the continous problem is singular. A possible alternative to reduce the error in these regions is to increase the number of elements in the partions where the error was considered large. The main issue is how to identify these regions, since the solution of the continuous problem is unknown. In this work we present the so-called residual estimate, which provides an error estimation approach which uses only the known values on the contours and the obtained approximation on a given discretization. We will discuss the relationship between the residual estimate and the error, and how to use the estimate for adaptively refining the mesh. Solutions for the Poisson equation and the Linear elasticity system of equations, and the residual estimates for the analysis of mesh refinement will be computed using the FreeFem++ software.

Many real-life applications such as the shape optimization of technological devices, the identification of parameters in environmental processes and flow control problems lead to optimization problems governed by systems of convection diusion partial dierential equations (PDEs). When convection dominates diusion, the solutions of these PDEs typically exhibit layers on small regions where the solution has large gradients. Hence, it requires special numerical techniques, which take into account the structure of the convection. The integration of discretization and optimization is important for the overall eciency of the solution process. Discontinuous Galerkin (DG) methods became recently as an alternative to the finite dierence, finite volume and continuous finite element methods for solving wave dominated problems like convection diusion equations since they possess higher accuracy. This thesis will focus on analysis and application of DG methods for linear-quadratic convection dominated optimal control problems. Because of the inconsistencies of the standard stabilized methods such as streamline upwind Petrov Galerkin (SUPG) on convection diusion optimal control problems, the discretize-then-optimize and the optimize-then-discretize do not commute. However, the upwind symmetric interior penalty Galerkin (SIPG) method leads to the same discrete optimality systems. The other DG methods such as nonsymmetric interior penalty Galerkin (NIPG) and incomplete interior penalty Galerkin (IIPG) method also yield the same discrete optimality systems when penalization constant is taken large enough. We will study a posteriori error estimates of the upwind SIPG method for the distributed unconstrained and control constrained optimal control problems. In convection dominated optimal control problems with boundary and/or interior layers, the oscillations are propagated downwind and upwind direction in the interior domain, due the opposite sign of convection terms in state and adjoint equations. Hence, we will use residual based a posteriori error estimators to reduce these oscillations around the boundary and/or interior layers. Finally, theoretical analysis will be confirmed by several numerical examples with and without control constraints

Projecting the ice sheets' contribution to sea-level rise is difficult because of the complexity of accurately modeling ice sheet dynamics for the full polar ice sheets, because of the uncertainty in key, unobservable parameters governing those dynamics, and because quantifying the uncertainty in projections is necessary when determining the confidence to place in them. This work presents the formulation and solution of the Bayesian inverse problem of inferring, from observations, a probability distribution for the basal sliding parameter field beneath the Antarctic ice sheet. The basal sliding parameter is used within a high-fidelity nonlinear Stokes model of ice sheet dynamics. This model maps the parameters "forward" onto a velocity field that is compared against observations. Due to the continental-scale of the model, both the parameter field and the state variables of the forward problem have a large number of degrees of freedom: we consider discretizations in which the parameter has more than 1 million degrees of freedom. The Bayesian inverse problem is thus to characterize an implicitly defined distribution in a high-dimensional space. This is a computationally demanding problem that requires scalable and efficient numerical methods be used throughout: in discretizing the forward model; in solving the resulting nonlinear equations; in solving the Bayesian inverse problem; and in propagating the uncertainty encoded in the posterior distribution of the inverse problem forward onto important quantities of interest. To address discretization, a hybrid parallel adaptive mesh refinement format is designed and implemented for ice sheets that is suited to the large width-to-height aspect ratios of the polar ice sheets. An efficient solver for the nonlinear Stokes equations is designed for high-order, stable, mixed finite-element discretizations on these adaptively refined meshes. A Gaussian approximation of the posterior distribution of parameters is defined, whose mean and covariance can be efficiently and scalably computed using adjoint-based methods from PDE-constrained optimization. Using a low-rank approximation of the covariance of this distribution, the covariance of the parameter is pushed forward onto quantities of interest.

In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.

A novel parallel block-based anisotropic adaptive mesh refinement (AMR) technique for multi-block body-fitted grids is proposed and described. Rather than adopting the more usual isotropic approach to mesh refinement, an anisotropic refinement procedure is proposed which allows refinement of grid blocks in each coordinate direction in an independent fashion. This allows for more efficient and accurate treatment of narrow layers and/or discontinuities which occur, for example, in the boundary and mixing layers of viscous flows, and in regions of strong non-linear wave interactions with shocks. The anisotropic AMR technique is implemented within an existing finite-volume framework, which encompasses both explicit and implicit solution methods, and is capable of performing calculations with second- and higher-order spatial accuracy. To clearly demonstrate the feasibility of the proposed technique, it is applied to the unsteady and steady-state solutions of both the advection diffusion equation, as well as the Euler equations, in two space dimensions.

In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.

A novel parallel block-based anisotropic adaptive mesh refinement (AMR) technique for multi-block body-fitted grids is proposed and described. Rather than adopting the more usual isotropic approach to mesh refinement, an anisotropic refinement procedure is proposed which allows refinement of grid blocks in each coordinate direction in an independent fashion. This allows for more efficient and accurate treatment of narrow layers and/or discontinuities which occur, for example, in the boundary and mixing layers of viscous flows, and in regions of strong non-linear wave interactions with shocks. The anisotropic AMR technique is implemented within an existing finite-volume framework, which encompasses both explicit and implicit solution methods, and is capable of performing calculations with second- and higher-order spatial accuracy. To clearly demonstrate the feasibility of the proposed technique, it is applied to the unsteady and steady-state solutions of both the advection diffusion equation, as well as the Euler equations, in two space dimensions.