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The purpose of this paper is to discuss the construction of a linear operator, referred to as the bubble transform, which maps scalar functions defined on into a collection of functions with local support. In fact, for a given simplicial triangulation of , the associated bubble transform produces a decomposition of functions on [Formula omitted] into a sum of functions with support on the corresponding macroelements. The transform is bounded in both and the Sobolev space , it is local, and it preserves the corresponding continuous piecewise polynomial spaces. As a consequence, this transform is a useful tool for constructing local projection operators into finite element spaces such that the appropriate operator norms are bounded independently of polynomial degree. The transform is basically constructed by two families of operators, local averaging operators and rational trace preserving cutoff operators., Author(s): Richard S. Falk[sup.1] , Ragnar Winther[sup.2] Author Affiliations: (1) Department of Mathematics, Rutgers University, 08854, Piscataway, NJ, USA (2) Department of Mathematics, University of Oslo, 0316, Oslo, Norway Introduction [...]

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the well-posedness of the continuous problem. The discretization methods we consider are finite element methods, in which a variational or weak formulation of the PDE problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace. After a brief introduction to finite element methods, we develop an abstract Hilbert space framework for analyzing the stability and convergence of such discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces, and stability is obtained by transferring Hodge-theoretic structures that ensure well-posedness of the continuous problem from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation self-contained for a variety of readers.

Byline: Ivar Astby (1), Ragnar Winther (2) Keywords: Stability; Granulopoiesis; Mathematical model Abstract: A class of mathematical models involving a convection-reaction partial differential equation (PDE) is introduced with reference to recovering human granulopoiesis after high dose chemotherapy with stem cell support. The stability properties of the model are addressed by means of numerical investigations and analysis. A simplified model with proliferation rate and mobilization rate independent of maturity shows that the model is stable as the maturation rate grows without bounds, but may go through stable and non-stable regimens as the maturation rate varies. It is also shown that the system is stable when parameters are chosen to approximate a real physiological situation. System characteristics do not change profoundly by introduction of a maturity-dependent proliferation and mobilization rate, as is necessary to make the model operate more in accordance with hematological observations. However, by changing the system mitotic responsiveness with respect to changes in cytokine level, the system is still stable but may show persistent oscillations much resembling clinical observations of cyclic neutropenia. Furthermore, in these cases, changes in the model feedback signal caused by, for instance, an impaired effective cytokine elimination by cell receptors may enforce these oscillations markedly. Author Affiliation: (1) Department of Informatics, University of Oslo, Box 1080, Blindern, 0316, Oslo, Norway (2) Centre of Mathematics for Applications (CMA) and Department of Informatics, 1080, Blindern, 0316, Oslo, Norway Article History: Registration Date: 01/01/2004 Received Date: 24/06/2003 Online Date: 05/07/2004

We construct a new Fortin operator for the lowest order Taylor-Hood element, which is uniformly stable both in $$L^2$$ and $$H^1$$. The construction, which is restricted to two space dimensions, is based on a tight connection between a subspace of the Taylor-Hood velocity space and the lowest order Nedelec edge element. General shape regular triangulations are allowed for the $$H^1$$-stability, while some mesh restrictions are imposed to obtain the $$L^2$$-stability. As a consequence of this construction, a uniform inf-sup condition associated the corresponding discretizations of a parameter dependent Stokes problem is obtained, and we are able to verify uniform bounds for a family of preconditioners for such problems, without relying on any extra regularity ensured by convexity of the domain. [ABSTRACT FROM AUTHOR]

This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for Partial Differential Equations in the summer of 2008. The main focus will be on an abstract approach to the construction of preconditioners for symmetric linear systems in a Hilbert space setting. Typical examples that are covered by this theory are systems of partial differential equations which correspond to saddle point problems. We will argue that the mapping properties of the coefficient operators suggest that block diagonal preconditioners are natural choices for these systems. To illustrate our approach a number of examples will be considered. In particular, parameter-dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several models which have previously been discussed in the literature. However, here each example is discussed with reference to a more unified abstract approach. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]