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. 1. Introduction. Research in computational mathematics is often motivated by the results of numerical experiments. Frequently, a result is conjectured based on behavior observed using a software environment such as MATLAB long before it is supported by any formal analytic results. In this sense, the "computational laboratory" plays just as important a role in modern mathematics as physical laboratories do in physics, chemistry, biology, and engineering. A similar observation applies in the context of teaching computational mathematics: carrying out investigative numerical experiments on particular topics helps students learn how to formulate hypotheses, design simple experiments to test them, and interpret the resulting data. As well as developing important deduction and interpretation skills, this hands-on approach is often more useful in helping students remember critical ideas over a significant period of time than a traditional textbook-only method.With this in mind, the Incompressible Flow & Iterative Solver Software (ifiss) toolbox [36] has been developed as a computational laboratory for the interactive
Abstract. 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.Key words. nullspace method, liquid crystals, saddle-point problems, unit-vector constraints AMS subject classifications. 65F08, 65F10, 65F50, 65H10, 65N22 DOI. 10.1137/120870219 1. Introduction. Many continuum models for the orientational properties of liquid crystals involve one or more state variables that are vector fields of unit length. The pointwise unit-vector constraints associated with discretizations of such models give rise to indefinite linear systems of saddle-point form, when these constraints are imposed via Lagrange multipliers. In problems such as these, indefiniteness also frequently manifests itself due to another influence (coupling with applied electric fields), and this leads to a double saddle-point structure. We are interested in the efficient numerical solution by iterative methods of large sparse linear systems of algebraic equations associated with such problems. We begin by presenting some background on these materials and models.
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