Alison Ramage

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case.

Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow

IFISS is a graphical Matlab package for the interactive numerical study of incompressible flow problems. It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions. The package can also be used as a computational laboratory for experimenting with state-of-the-art preconditioned iterative solvers for the discrete linear equation systems that arise in incompressible flow modelling. A unique feature of the package is its comprehensive nature; for each problem addressed, it enables the study of both discretization and iterative solution algorithms as well as the interaction between the two and the resulting effect on overall efficiency.

Implicit solution of hyperbolic equations with space-time adaptivity

Adaptivity in space and time is introduced to control the error in the numerical solution of hyperbolic partial differential equations. The equations are discretised by a finite volume method in space and an implicit linear multistep method in time. The computational grid is refined in blocks. At the boundaries of the blocks, there may be jumps in the step size. Special treatment is needed there to ensure second order accuracy and stability. The local truncation error of the discretisation is estimated and is controlled by changing the step size and the time step. The global error is obtained by integration of the error equations. In the implicit scheme, the system of linear equations at each time step is solved iteratively by the GMRES method. Numerical examples executed on a parallel computer illustrate the method.

IFISS : a computational laboratory for investigating incompressible flow problems

The Incompressible Flow & Iterative Solver Software (\ifiss) package contains software which can be run with MATLAB or Octave to create a computational laboratory for the interactive numerical study of incompressible flow problems. It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions, together with state-of-the-art preconditioned iterative solvers for the resulting discrete linear equation systems. In this paper we give a flavor of the codes main features and illustrate its applicability using several case studies. We aim to show that \ifiss can be a valuable tool in both teaching and research.

An Analysis of Smoothing Effects of Upwinding Strategies for the Convection-Diffusion Equation

Using a technique for constructing analytic expressions for discrete solutions to the convection-diffusion equation, we examine and characterize the effects of upwinding strategies on solution quality. In particular, for grid-aligned flow and discretization based on bilinear finite elements with streamline upwinding, we show precisely how the amount of upwinding included in the discrete operator affects solution oscillations and accuracy when different types of boundary layers are present. This analysis provides a basis for choosing a streamline upwinding parameter which also gives accurate solutions for problems with non-grid-aligned and variable speed flows. In addition, we show that the same analytic techniques provide insight into other discretizations, such as a finite difference method that incorporates streamline diffusion and the isotropic artificial diffusion method.

A Preconditioned Nullspace Method for Liquid Crystal Director Modeling

We present a preconditioned nullspace method for the numerical solution of large sparse linear systems that arise from discretizations of continuum models for the orientational properties of liquid crystals. The approach effectively deals with pointwise unit-vector constraints, which are prevalent in such models. The indefinite, saddle-point nature of such problems, which can arise from either or both of two sources (pointwise unit-vector constraints, coupled electric fields), is illustrated. Both analytical and numerical results are given for a model problem.

Preconditioning for Radial Basis Function Partition of Unity Methods

Meshfree radial basis function (RBF) methods are of interest for solving partial differential equations due to attractive convergence properties, flexibility with respect to geometry, and ease of implementation. For global RBF methods, the computational cost grows rapidly with dimension and problem size, so localised approaches, such as partition of unity or stencil based RBF methods, are currently being developed. An RBF partition of unity method (RBF–PUM) approximates functions through a combination of local RBF approximations. The linear systems that arise are locally unstructured, but with a global structure due to the partitioning of the domain. Due to the sparsity of the matrices, for large scale problems, iterative solution methods are needed both for computational reasons and to reduce memory requirements. In this paper we implement and test different algebraic preconditioning strategies based on the structure of the matrix in combination with incomplete factorisations. We compare their performance for different orderings and problem settings and find that a no-fill incomplete factorisation of the central band of the original discretisation matrix provides a robust and efficient preconditioner.

A characterisation of oscillations in the discrete two-dimensional convection-diffusion equation

It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh Peclet number is large. The oscillatory functions are determined as solutions to a set of three-term recurrences, and the weights are determined by the boundary conditions. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh Peclet number and boundary conditions of the problem.

Efficient Moving Mesh Methods for

This paper describes a robust and efficient numerical scheme for solving the system of six coupled partial differential equations which arises when using

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