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.

Adaptive anisotropic refinement of finite element meshes allows to reduce the computational effort required to achieve a specified accuracy of the solution of a PDE problem. We present a new approach to adaptive refinement and demonstrate that this allows to construct algorithms which generate very flexible and efficient anisotropically refined meshes, even improving the convergence order compared to adaptive isotropic refinement if the problem permits.

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.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / The mathematical modeling of multiphase flows involves the interaction between deformable and moving geometries with the fluid in which they are dispersed (immersed). This kind of interaction is present in many practical applications. A common approach to handle these problems is the so called Front-Tracking/Front-Capturing Hybrid Methods. This methodology consists in separating the problem into two domains: an eulerian domain, which is kept fixed and is used to discretize the fluid equations of both phases, and a lagrangian domain, which is used to solve the equations of motion of the interface. Since there is no geometric dependence between these two domains, the method can easily handle moving and deformable interfaces that are dispersed in the flow. Following this line of research, the present work aims to capture accurately details of such flows by resolving adequately the relevant physical scales in time and in space. This can be achieved by applying locally refined meshes which adapt dynamically to cover special flow regions, e.g. the vicinity of the fluid-fluid interfaces. To obtain the required resolution in time, a semi-implicit second order discretization to solve the Navier-Stokes equations is used. The turbulence modeling is introduced in the present work through Large Eddy Simulation. The eficiency and robustness of the methodology applied are verified via convergence analysis, as well as with simulations of one-phase and two-phase flows for several Reynolds numbers. The results of two-phase flows, with one bubble and with multiple bubbles, are presented. The results obtained for a single bubble case are compared with Clift's shape diagram (Clift et al., 1978). / A modelagem matemática de escoamentos multifásicos envolve a interação de geometrias móveis e deformáveis com o meio fluido que as envolve. Este tipo de interação faz parte de uma extensa lista de aplicações. Uma linha proposta para o tratamento num érico deste tipo de problema são os métodos híbridos Front-Tracking/Front-Capturing. Esta abordagem leva à separação do problema em dois domínios distintos (líquido/gás e líquido/líquido), um domínio fixo, euleriano, utilizado para discretizar as equações de ambas as fases, e outro móvel, lagrangiano, usado para as interfaces. Para o presente trabalho, na metologia utilizada, ambos os domínios são geometricamente independentes e não apresentam restrição quanto ao movimento e à deformação da fase dispersa. Seguindo esta linha, no presente trabalho propõe-se capturar detalhes deste de tipo escoamento, resolvendo adequadamente as escalas físicas do tempo e do espaço, utilizando malhas bloco estruturada refinadas localmente, as quais se adaptam dinamicamente para recobrir as regiões de interesse do escoamento (como, por exemplo, ao redor da interface fluido-fluido). Para se obter a resolução necessária no tempo, é usada uma discretização semi-implícita de segunda ordem para solucionar as equações de Navier-Stokes. A modelagem da turbulência é introduzida no presente trabalho via Simulação de Grandes Escalas. A eficiência e a robustez da metodologia implementada são verificadas via análise de convergência do método, bem como a simulação de escoamentos monofásicos e bifásicos para diferentes números Reynolds. São também apresentados resultados para escoamentos bifásicos com uma só bolha assim como para múltiplas bolhas. Os resultados de escoamentos mono-bolhas são comparados com o diagrama de forma de Clift et al. (1978). / Doutor em Engenharia Mecânica

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.

Este trabalho apresenta um estudo sobre a solução numérica da equação geral de advecção-difusão usando uma metodologia numérica conservativa. Para a discretização espacial, é usado o Método de Volumes Finitos devido à natureza conservativa da equação em questão. O método é configurado de modo a ter suas variáveis centradas em centro de célula e, para as variáveis, como a velocidade, centradas nas faces um método de interpolação de segunda ordem é utilizado para um ajuste numérico ao centro. Embora a implementação computacional tenha sido feita de forma paramétrica de maneira a acomodar outros esquemas numéricos, a discretização temporal dá ênfase ao Método de Crank-Nicolson. Tal método numérico, sendo ele implícito, dá origem a um sistema linear de equações que, aqui, é resolvido empregando-se o Método Multigrid-Multinível. A corretude do código implementado é verificada a partir de testes por soluções manufaturadas, de modo a checar se a ordem de convergência prevista em teoria é alcançada pelos métodos numéricos. Um jato laminar é simulado, com o acoplamento entre a equação de Navier-Stokes e a equação geral de advecção-difusão, em um domínio computacional tridimensional. O jato é uma forma de vericar se o algoritmo de geração de malhas adaptativas funciona corretamente. O módulo produzido neste trabalho é baseado no código computacional AMR3D-P desenvolvido pelos grupos de pesquisa do IME-USP e o MFLab/FEMEC-UFU (Laboratório de Dinâmica de Fluidos da Universidade Federal de Uberlândia). A linguagem FORTRAN é utilizada para o desenvolvimento da metodologia numérica e as simulações foram executadas nos computadores do LabMAP(Laboratório da Matemática Aplicada do IME-USP) e do MFLab/FEMEC-UFU. / This work presents a study about the numerical solution of variable coecients advectiondi usion equation, or simply, general advection-diusion equation using a conservative numerical methodology. The Finite Volume Method is choosen as discretisation of the spatial domain because the conservative nature of the focused equation. This method is set up to have the scalar variable in a cell centered scheme and the vector quantities, such velocity, are face centered and they need a second order interpolation to get adjusted to the cell center. The computational code is parametric, in which, any implicit temporal discretisation can be choosen, but the emphasis relies on Crank-Nicolson method, a well-known second order method. The implicit nature of aforementioned method gives a linear system of equations which is solved here by the Multilevel-Multigrid method. The correctness of the computational code is checked by manufactured solution method used to inspect if the theoretical order of convergence is attained by the numerical methods. A laminar jet is simulated, coupling the Navier-Stokes equation and the general advection-diusion equation in a 3D computational domain. The jet is a good way to check the corectness of adaptative mesh renement algorithm. The module designed here is based in a previous implemented code AMR3D-P designed by IME-USP and MFLab/FEMEC-UFU (Fluid Dynamics Laboratory, Federal University of Uberlândia). The programming language used is FORTRAN and the simulations were run in LabMAP(Applied Mathematics Laboratoy at IME-USP) and MFLab/FEMEC-UFU computers.

The numerical solution of the convection-diffusion-reaction problem is considered in two and three dimensions. A stabilized finite element method of Galerkin/Least squares type accomodates diffusion-dominated as well as convection- and/or reaction- dominated situations. The resolution of boundary layers occuring in the singularly perturbed case is accomplished using anisotropic mesh refinement in boundary layer regions. In this paper, the standard analysis of the stabilized Galerkin method on isotropic meshes is extended to more general meshes with boundary layer refinement. Simplicial Lagrangian elements of arbitrary order are used.