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

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 discretization by mixed finite element methods and posteriori error estimation computed solutions, together state-of-the-art preconditioned iterative solvers resulting discrete linear equation systems. In this paper we give flavor code's...

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

Abstract The linear system arising from a Lagrange‐Galerkin mixed finite element approximation of the Navier–Stokes and continuity equations is symmetric indefinite has same block structure as discretization Stokes problem. This paper considers iterative solution such system, comparing performance one‐level preconditioned conjugate residual method for matrices with that more traditional two‐level pressure correction approach. Asymptotic estimates amount work involved in each are given...

Using a technique for constructing analytic expressions discrete solutions to the convection-diffusion equation, we examine and characterize effects of upwinding strategies on solution quality. In particular, grid-aligned flow discretization based bilinear finite elements with streamline upwinding, show precisely how amount included in operator affects oscillations accuracy when different types boundary layers are present. This analysis provides basis choosing parameter which also gives...

This paper describes a robust and efficient numerical scheme for solving the system of six coupled partial differential equations which arises when using $Q$-tensor theory to model behavior nematic liquid crystal cell under influence an applied electric field. The key novel feature is use full moving mesh equation approach generate adaptive accurately resolves important solution features. includes new monitor function based on local measure biaxiality. In addition, time-step control used...

It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by discretisation. However, except for one-dimensional problems, there little analysis of this phenomenon. In paper, we an two-dimensional problem with constant flow aligned grid, based on a Fourier decomposition solution. For Galerkin bilinear finite element discretisations, derive closed form expressions coefficients, showing them...

Abstract Iterative solvers are widely regarded as the most efficient way to solve very large linear systems arising from finite element models. Their memory requirements small compared those for direct solvers. Consequently, there is a major interest in iterative methods and particularly preconditioning necessary achieve rapid convergence. In this paper we present new element‐based preconditioners specifically designed elasticity elasto‐plastic problems. The study presented here restricted...

We consider the nonlinear systems of equations that result from discretizations a prototype variational model for equilibrium director field characterizing orientational properties liquid crystal material. In presence pointwise unit-vector constraints and coupled electric fields, numerical solution such by Lagrange--Newton methods leads to linear with double saddle-point form, which we have previously proposed preconditioned nullspace method as an effective solver [A. Ramage E. C. Gartland,...

This article illustrates the use of moving mesh methods for solving partial differential equation (PDE) problems in Q-tensor theory liquid crystals. We present results an initial study using a simple one-dimensional test problem which feasibility applying adaptive grid techniques such situations. describe how grids are computed equidistribution principle, and investigate comparative accuracy uniform strategies, both theoretically via numerical examples.

In this paper we illustrate the suitability of an adaptive moving mesh method for modelling a one‐dimensional liquid crystal cell using Q‐tensor theory. Specifically, consider time‐dependent problem in Pi‐cell geometry which admits two topologically different equilibrium states and model order reconstruction occurs on application electric field. An finite element grid is used where points are moved according to equidistribution monitor function based specific property Q‐tensor. We show that...

Due to recent advances in fast iterative solvers the field of computational fluid dynamics, more complex problems which were previously beyond scope standard techniques can be tackled. In this paper, we describe one such situation, namely, modelling interaction flow and molecular orientation a as liquid crystal. Specifically, consider nematic crystal spatially inhomogeneous situation where orientational order is described by second rank alignment tensor. The evolution determined two coupled...

Preconditioning methods are widely used in conjunction with the conjugate gradient method for solving large sparse symmetric linear systems arising from discretisation of selfadjoint elliptic partial differential equations. Many different preconditioners have been proposed, and they generally analysed compared using model problems: simple discretisations Laplacian operators on regular computational grids, two space dimensions. For such problems there highly competitive multigrid methods, it...

The properties of liquid crystals can be modelled using an order parameter which describes the variability local orientation rod-like molecules. Defects in director field arise due to external factors such as applied electric or magnetic fields, constraining geometry cell containing crystal material. Understanding formation and dynamics defects is important design control devices, poses significant challenges for numerical modelling. In this paper we consider solution a Q-tensor model...

Use of data assimilation techniques is becoming increasingly common across many application areas. The inverse Hessian (and its square root) plays an important role in several different aspects these processes. In geophysical and engineering applications, the Hessian-vector product typically defined by sequential solution a tangent linear adjoint problem; for Hessian, however, no such definition possible. Frequently, requirement to work matrix-free environment means that compact...