Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type

In this work, we consider parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle, in the simplest case that the diffusion parameter is the same for all equations of the system. The solution is approximated on a Shishkin mesh with two splitting or additive methods in time and standard central differences in space. It is proved that they are first-order in time and almost second-order in space uniformly convergent schemes. The additive schemes decouple the components of the vector solution at each time level of the discretization which makes the computation more efficient. Moreover, a multigrid algorithm is used to solve the resulting linear systems. Numerical results for some test problems are showed, which illustrate the theoretical results and the efficiency of the splitting and multigrid techniques

Geometric Multigrid Methods for Flow Problems in Highly Heterogeneous Porous Media

In this dissertation, we develop geometric multigrid methods for the finite element approximation of flow problems (e:g:, Stokes, Darcy and Brinkman models) in highly heterogeneous porous media. Our method is based on H^(div)-conforming discontinuous Galerkin methods and the Arnold-Falk-Winther (AFW) type smoothers. The main advantage of using H^(div)-conforming elements is that the discrete velocity field will be globally divergence-free for incompressible fluid flows. Besides, the smoothers used are of overlapping domain decomposition Schwarz type and employ a local Helmholtz decomposition. Our flow solvers are the combination of multigrid preconditioners and classical iterative solvers. The proposed preconditioners act on the combined velocity and pressure space and thus does not need a Schur complement approximation. There are two main ingredients of our preconditioner: first, the AFW type smoothers can capture a meaningful basis on local divergence free subspace associated with each overlapping patch; second, the grid operator does not increase the divergence from the coarse divergence free subspace to the fine one as the divergence free spaces are nested. We present the convergence analysis for the Stokes' equations and Brinkman's equations ( with constant permeability field ), as well as extensive numerical experiments. Some of the numerical experiments are given to support the theoretical results. Even though we do not have analysis work for the highly heterogeneous and highly porous media cases, numerical evidence exhibits strong robustness, efficiency and unification of our algorithm

Comparative Study of Uniform and Graded Meshes for Solving Convection-Diffusion Equation with Quadratic Source

Due to its fundamental nature, the problems of convection-diffusion are discussed in various aviation, science and engineering applications. Among major applications are in the study of the dynamics of aircraft wake vortex and its interaction with turbulent jet which is a very serious hazard in aviation. Other applications include those in the investigation of intrusive sampling of jet engine exhaust gases, and the effectiveness of hot fluid injection in the removal of ice on aircraft wings. The numerical solutions of convection-diffusion require proper meshing schemes. Among major meshes in computational fluid dynamics are those of uniform, piecewise-uniform, graded, and hybrid over which the solutions of discretized governing equations are found. Bad solutions as spurious fluctuations, over- or under-predictions, and excessive computation time might be the results of unwitting application of the meshes. Accentuating comparative effectiveness of two meshes, namely uniform mesh and graded mesh with mesh expansion factor, this paper takes the solution of a convection-diffusion equation with quadratic source term into account. The problem is solved by assigning several values of mesh expansion factor to graded mesh, while mesh number is kept constant. The factors are calculated based on the generalization of their logarithmically linear relationship with low Peclet numbers derived in previous work. Based on the values of Peclet number, five test cases are considered. Graded mesh is proven relatively more robust, particularly due the solution on the mesh being free from spurious fluctuation. Furthermore, the accuracy level of the solution of up to two order of magnitude higher is obtained. The mesh expansion factor therefore contributes to a stable and highly accurate solution corresponding to all interested Peclet numbers

Positivity-preserving multigrid and multilevel methods

Multigrids methods are extremely effective algorithms for solving the linear systems that arise from discretization of many differential equations in computational mathematics. A multigrid method uses a hierarchy of representations of the problem to achieve its efficiency and fast convergence. Originally inspired by a problem in adaptive mesh generation, this thesis focuses on the application of multigrid methods to a range of problems where the solution is required to preserve some additional properties during the iteration process. The major contribution of this thesis is the development of multigrid methods with the additional feature of preserving solution positivity: We have formulated both a multiplicative form multigrid method and a modified unigrid algorithm with constraints that are able to preserve positivity of the approximate solution at every iteration while maintaining convergence properties typical of normal multigrid methods. We have applied these algorithms to the 1D adaptive mesh generation problem to guarantee mesh nonsingularity, to singularly perturbed semilinear reaction-diffusion equations to compute unstable solutions, and to nonlinear diffusion equations. Numerical results show that our algorithms are effective and also possess good convergence properties

Space-time discontinuous Petrov-Galerkin finite elements for transient fluid mechanics

Initial mesh design for computational fluid dynamics can be a time-consuming and expensive process. The stability properties and nonlinear convergence of most numerical methods rely on a minimum level of mesh resolution. This means that unless the initial computational mesh is fine enough, convergence can not be guaranteed. Any meshes below this minimum resolution level are termed to be in the ``pre-asymptotic regime.'' This condition implies that meshes need to in some way anticipate the solution before it is known. On top of the minimum requirement that the surface meshes must adequately represent the geometry of the problem under consideration, resolution requirements on the volume mesh make the CFD practitioner's job significantly more time consuming. In contrast to most other numerical methods, the discontinuous Petrov-Galerkin finite element method retains exceptional stability on extremely coarse meshes. DPG is also inherently very adaptive. It is possible to compute the residual error without knowledge of the exact solution, which can be used to robustly drive adaptivity. This results in a very automated technology, as the user can initialize a computation on the coarsest mesh which adequately represents the geometry then step back and let the program solve and adapt iteratively until it resolves the solution features. A common complaint of minimum residual methods by computational fluid dynamics practitioners is that they are not locally conservative. In this thesis, this concern is addressed by developing a locally conservative DPG formulation by augmenting the system with Lagrange multipliers. The resulting DPG formulation is then proved to be robust and shown to produce superior numerical results over standard DPG on a selection of test problems. Adaptive convergence to steady incompressible and compressible Navier-Stokes solutions was explored in Jesse Chan's and Nathan Roberts' dissertations. Space-time offers a natural extension to transient problems as it preserves the stability and adaptivity properties of DPG in the time dimension. Space-time also offers more extensive parallelization capability than problems treated with traditional time stepping as it allows multigrid concurrently in both space and time. A proof of concept space-time DPG formulation is developed for transient convection-diffusion. The robust test norms derived for steady convection-diffusion are extended to the space-time case and proofs of robustness are provided. Numerical results verify the robust behavior and near $L^2$ optimality of the resulting solutions. The space-time formulation for convection-diffusion is then extended to transient incompressible and compressible Navier-Stokes by analogy. Several numerical experiments are performed, but a mathematical analysis is not attempted for these nonlinear problems. Several side topics are explored such as a study of the compressible Navier-Stokes equations under various variable transformations and the development of consistent test norms through the concept of physical entropy.Computational Science, Engineering, and Mathematic

Final report on the Copper Mountain conference on multigrid methods

The Copper Mountain Conference on Multigrid Methods was held on April 6-11, 1997. It took the same format used in the previous Copper Mountain Conferences on Multigrid Method conferences. Over 87 mathematicians from all over the world attended the meeting. 56 half-hour talks on current research topics were presented. Talks with similar content were organized into sessions. Session topics included: fluids; domain decomposition; iterative methods; basics; adaptive methods; non-linear filtering; CFD; applications; transport; algebraic solvers; supercomputing; and student paper winners