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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. [PUBLICATION ABSTRACT]

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 code's 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.

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.

Considering saddle-point systems of the Karush–Kuhn–Tucker (KKT) form, we propose approximations of the “ideal” block diagonal preconditioner based on the exact Schur complement proposed by Murphy et al. (SIAM J Sci Comput 21(6):1969–1972, 2000). We focus on the case where the (1,1) block is symmetric and positive definite, but with a few very small eigenvalues that possibly affect the convergence of Krylov subspace methods like Minres. Assuming that these eigenvalues and their associated eigenvectors are available, we first propose a Schur complement preconditioner based on this knowledge and establish lower and upper bounds on the preconditioned Schur complement. We next analyse theoretically the spectral properties of the preconditioned KKT systems using this Schur complement approximation in two spectral preconditioners of block diagonal forms. In addition, we derive a condensed “two in one” formulation of the proposed preconditioners in combination with a preliminary level of preconditioning on the KKT system. Finally, we illustrate on a PDE test case how, in the context of a geometric multigrid framework, it is possible to construct practical block preconditioners that help to improve on the convergence of Minres.

We consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material. In the presence of pointwise unit-vector constraints and coupled electric fields, the numerical solution of such equations by Lagrange–Newton methods leads to linear systems with a double saddle-point form, for which we have previously proposed a preconditioned nullspace method as an effective solver [A. Ramage and E. C. Gartland, Jr., SIAM J. Sci. Comput., 35 (2013), pp. B226–B247]. Here we propose and analyze a modified outer iteration (\"renormalized Newton method\") in which the orientation variables are normalized onto the constraint manifold at each iterative step. This scheme takes advantage of the special structure of these problems, and we prove that it is locally quadratically convergent. The renormalized Newton method bears some resemblance to the truncated Newton method of computational micromagnetics, and we compare and contrast the two. This brings to light some anomalies of the truncated Newton method.

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 Péclet 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 Péclet number and boundary conditions of the problem.

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.

Preconditioning methods are widely used in conjunction with the conjugate gradient method for solving large sparse symmetric linear systems arising from the discretisation of selfadjoint linear elliptic partial differential equations. Many different preconditioners have been proposed, and they are generally analysed and compared using model problems: simple discretisations of Laplacian operators on regular computational grids, generally in two space dimensions. For such model problems there are highly competitive multigrid methods, and it is principally for geometrically irregular (nonmodel) problems that the applicability and economy of preconditioned conjugate gradient methods are most useful. This is particularly true for problems on irregular unstructured three-dimensional grids. This paper is concerned with the comparison of preconditioners for finite element discretisations of three-dimensional selfadjoint elliptic problems on irregular and unstructured computational grids. It is argued that simple preconditioners, which are inferior for regular grid problems in two dimensions, are competitive for irregular grid problems in three dimensions.

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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.