Abstract. In this paper, proper orthogonal decomposition (POD) is used for model reduction of mixed finite element (MFE) for the nonstationary Navier-Stokes equations and error estimates between a reference solution and the POD solution of reduced MFE formulation are derived. The basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual MFE method for the nonstationary Navier-Stokes equations or from physics system trajectories by drawing samples of experiments and interpolation (or data assimilation), and then the basis functions of the usual MFE method are substituted with the POD basis functions reconstructed by the elements of the ensemble to derive the POD-reduced MFE formulation for the nonstationary Navier-Stokes equations. It is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between POD solution of reduced MFE formulation and the reference solution is consistent with theoretical results. Moreover, it is also shown that this result validates the feasibility and efficiency of the POD method.Key words. mixed finite element method, proper orthogonal decomposition, the nonstationary Navier-Stokes equations, error estimate AMS subject classifications. 65N30, 35Q10DOI. 10.1137/070689498 1. Introduction. The mixed finite element (MFE) method is one of the important approaches for solving systems of partial differential equations, for example, the nonstationary Navier-Stokes equations (see [1], [2], or [3]). However, the computational model for the fully discrete system of MFE solutions of the nonstationary Navier-Stokes equations yields very large systems that are computationally intensive. Thus, an important problem is how to simplify the computational load and save time-consuming calculations and resource demands in the actual computational process in a way that guarantees a sufficiently accurate and efficient numerical solution. Proper orthogonal decomposition (POD), also known as Karhunen-Loève expansions in signal analysis and pattern recognition (see [4]), or principal component analysis in statistics (see [5]), or the method of empirical orthogonal functions in geophysical fluid dynamics (see [6], [7]) or meteorology (see [8]), is a technique offering adequate approximation for representing fluid flow with reduced number of degrees of freedom, i.e., with lower dimensional models (see [9]), so as to alleviate the computational load

A novel family of RGD-containing molecule (Tablysin-15) has been molecularly characterized from the salivary gland of the hematophagous horsefly Tabanus yao. Tablysin-15 does not share primary sequence homology to any disintegrin discovered so far, and displays an RGD motif in the N-terminus of the molecule. It is also distinct from disintegrins from Viperidae since its mature form is not released from a metalloproteinase precursor. Tablysin-15 exhibits high affinity for platelet αIIbβ3 and endothelial cell αvβ3 integrins, but not for α5β1 or α2β1. Accordingly, it blocks endothelial cell adhesion to vitronectin (IC50 ~ 1 nM) and marginally to fibronectin (IC50 ~ 1 µM), but not to collagen. It also inhibits FGF-induced endothelial cell proliferation, and attenuates tube formation in vitro. In platelets, Tablysin-15 inhibits aggregation induced by collagen, ADP and convulxin, and prevents static platelet adhesion to immobilized fibrinogen. In addition, solid-phase assays and flow cytometry demonstrates that αIIbβ3 binds to Tablysin-15. Moreover, immobilized Tablysin-15 supports platelet adhesion by a mechanism which was blocked by anti-integrin αIIbβ3 monoclonal antibody (e.g. abciximab) or by EDTA. Furthermore, Tablysin-15 dose-dependently attenuates thrombus formation to collagen under flow, without affecting platelet adhesion to collagen fibrils. Consistent with these findings, Tablysin-15 displays antithrombotic properties in vivo suggesting that it is a useful tool to block αIIbβ3, or as a prototype to develop antithrombotics. The RGD motif in the unique sequence of Tablysin-15 represents a novel template for studying the structure-function relationship of the disintegrin family of inhibitors.

A proper orthogonal decomposition (POD) method is applied to a usual finite element (FE) formulation for parabolic equations so that it is reduced into a POD FE formulation with lower dimensions and enough high accuracy. The errors between the reduced POD FE solution and the usual FE solution are analyzed. It is shown by numerical examples that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is also shown that this validates the feasibility and efficiency of POD method.

We have first identified and characterized two IgE-binding proteins, Tab a 1, an Ag 5-like protein and Tab a 2, a hyaluronidase, from the horsefly salivary glands. They appear to be of importance for the allergic reactions induced by horsefly bite. These allergens are thus not only found in stinging but also found in hematophagous insects. These results also provided support for the presence of the so-called wasp-horsefly syndrome (WHS).