Incompressible flows -- flows in which variations in the density of a fluid are negligible -- arise in a wide variety of applications, from hydraulics to aerodynamics. The incompressible Navier-Stokes equations which govern such flows are also of fundamental physical and mathematical interest. They are believed to hold the key to understanding turbulent phenomena; precise conditions for the existence and uniqueness of solutions remain unknown -- and establishing such conditions is the subject of one of the Clay Mathematics Institute's Millennium Prize Problems. Typical solutions of incompressible flow problems involve both fine- and large-scale phenomena, so that a uniform finite element mesh of sufficient granularity will at best be wasteful of computational resources, and at worst be infeasible because of resource limitations. Thus adaptive mesh refinements are required. In industry, the adaptivity schemes used are ad hoc, requiring a domain expert to predict features of the solution. A badly chosen mesh may cause the code to take considerably longer to converge, or fail to converge altogether. Typically, the Navier-Stokes solve will be just one component in an optimization loop, which means that any failure requiring human intervention is costly. Therefore, I pursue technological foundations for a solver of the incompressible Navier-Stokes equations that provides robust adaptivity starting with a coarse mesh. By robust, I mean both that the solver always converges to a solution in predictable time, and that the adaptive scheme is independent of the problem -- no special expertise is required for adaptivity. The cornerstone of my approach is the discontinuous Petrov-Galerkin (DPG) finite element methodology developed by Leszek Demkowicz and Jay Gopalakrishnan. For a large class of problems, DPG can be shown to converge at optimal rates. DPG also provides an accurate mechanism for measuring the error, and this can be used to drive adaptive mesh refinements. Several approximations to Navier-Stokes are of interest, and I study each of these in turn, culminating in the study of the steady 2D incompressible Navier-Stokes equations. The Stokes equations can be obtained by neglecting the convective term; these are accurate for "creeping" viscous flows. The Oseen equations replace the convective term, which is nonlinear, with a linear approximation. The steady-state incompressible Navier-Stokes equations approximate the transient equations by neglecting time variations. Crucial to this work is Camellia, a toolbox I developed for solving DPG problems which uses the Trilinos numerical libraries. Camellia supports 2D meshes of triangles and quads of variable polynomial order, allows simple specification of variational forms, supports h- and p-refinements, and distributes the computation of the stiffness matrix, among other features. The central contribution of this dissertation is design and development of mathematical techniques and software, based on the DPG method, for solving the 2D incompressible Navier-Stokes equations in the laminar regime (Reynolds numbers up to about 1000). Along the way, I investigate approximations to these equations -- the Stokes equations and the Oseen equations -- followed by the steady-state Navier-Stokes equations. / text

In the last half a century, Computational Fluid Dynamics (CFD) has been established as an important complementary part and some times a significant alternative to Experimental and Theoretical Fluid Dynamics. Development of efficient computational algorithms for digital simulation of fluid flows has been an ongoing research effort in CFD. An accurate numerical simulation of compressible Euler equations, which are the gov-erning equations of high speed flows, is important in many engineering applications like designing of aerospace vehicles and their components. Due to nonlinear nature of governing equations, such flows admit solutions involving discontinuities like shock waves and contact discontinuities. Hence, it is nontrivial to capture all these essential features of the flows numerically. There are various numerical methods available in the literature, the popular ones among them being the Finite Volume Method (FVM), Finite Difference Method (FDM), Finite Element Method (FEM) and Spectral method. Kinetic theory based algorithms for solving Euler equations are quite popular in finite volume framework due to their ability to connect Boltzmann equation with Euler equations. In kinetic framework, instead of dealing directly with nonlinear partial differential equations one needs to deal with a simple linear partial differential equation. Recently, FEM has emerged as a significant alternative to FVM because it can handle complex geometries with ease and unlike in FVM, achieving higher order accuracy is easier. High speed flows governed by compressible Euler equations are hyperbolic partial differential equations which are characterized by preferred directions for information propagation. Such flows can not be solved using traditional FEM methods and hence, stabilized methods are typically introduced. Various stabilized finite element methods are available in the literature like Streamlined-Upwind Petrov Galerkin (SUPG) method, Galerkin-Least Squares (GLS) method, Taylor-Galerkin method, Characteristic Galerkin method and Discontinuous Galerkin Method. In this thesis a novel stabilized finite element method called as Kinetic Streamlined-Upwind Petrov Galerkin (KSUPG) method is formulated. Both explicit and implicit versions of KSUPG scheme are presented. Spectral stability analysis is done for explicit KSUPG scheme to obtain the stable time step. The advantage of proposed scheme is, unlike in SUPG scheme, diffusion vectors are obtained directly from weak KSUPG formulation. The expression for intrinsic time scale is directly obtained in KSUPG framework. The accuracy and robustness of the proposed scheme is demonstrated by solving various test cases for hyperbolic partial differential equations like Euler equations and inviscid Burgers equation. In the KSUPG scheme, diffusion terms involve computationally expensive error and exponential functions. To decrease the computational cost, two variants of KSUPG scheme, namely, Peculiar Velocity based KSUPG (PV-KSUPG) scheme and Circular distribution based KSUPG (C-KSUPG) scheme are formulated. The PV-KSUPG scheme is based on peculiar velocity based splitting which, upon taking moments, recovers a convection-pressure splitting type algorithm at the macroscopic level. Both explicit and implicit versions of PV-KSUPG scheme are presented. Unlike KSUPG and PV-KUPG schemes where Maxwellian distribution function is used, the C-KUSPG scheme uses a simpler circular distribution function instead of a Maxwellian distribution function. Apart from being computationally less expensive it is less diffusive than KSUPG scheme.