Summary. In this paper we derive error estimates for a class of finite element approximation of the Stokes equation. These elements, popular among engineers, are conforming lagrangian both in velocity and pressure and therefore based on a mixed variational principle. The error estimates are established from a new Brezzi-type inequality for this kind of mixed formulation. The results are true in 2 or 3 dimensions.Subject Classifications: AMS: 65N30; CR: 5.13, 5.17. O. IntroductionIn general, numerical simulation of flows involves complex geometries. The finite element method (F.E.M.) is a natural tool of discretization in such cases. Error estimates for the Stokes F.E.M. approximation yield identical estimates for the Navier-Stokes equations, under suitable existence and uniqueness hypotheses, see Jamet-Raviart [5] and Le Tallec [7].Among the first methods proposed for the F.E.M. approximation of these problems, the mixed method of Hood-Taylor [3] is widely used in engineering. The most popular element is a conforming isoparametric quadrilateral with 9-node biquadratic interpolation for the velocity u and 4-node bilinear for the pressure p. A variant of the same method uses triangular elements P2 in u and P1 in p. (see and Hood-Taylor [3]).We show that both approximations have H 1 error estimates of order h z for u and h for p. Therefore these approximations are optimal.The error analysis shows that an element of lower degree in u can also be used, namely: a bilinear (resp. linear) approximation for u and a bilinear (resp. linear) for p, where each quadrilateral (resp. triangle) for p is actually a rnacroelement made of four quadrilaterals (resp. triangles) for n, see Fig. 1.

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Many textiles do not noticeably stretch under their own weight. Unfortunately, for better performance many cloth solvers disregard this fact. We propose a method to obtain very low strain along the warp and weft direction using Constrained Lagrangian Mechanics and a novel fast projection method. The resulting algorithm acts as a velocity filter that easily integrates into existing simulation code.

We study the following problem: given a polynomial order of approximation n and the corresponding Bézier tensor product patches over an unstructured quadrilateral mesh made of convex quadrilaterals with vertices of any valence , is there a solution to the G 1 ( and as a consequence the C 1 ) approximation (resp. interpolation ) problem ? To illustrate the interpolation case , constraints defining regularity conditions across patches have to be satisfied. The resulting number of free degrees of freedom must be such that the interpolation problem has a solution! This is similar to studying the minimal determining set (MDS) for a C 1 continuity construction. *

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