Gabriel Acosta

An optimal Poincaré inequality in ¹ for convex domains

For convex domains Ω C R n with diameter d we prove ∥u∥ L 1 (ω) ≤ d 2 ∥⊇ u ∥ L 1 (ω) for any u with zero mean value on w. We also show that the constant 1/2 in this inequality is optimal.

An adaptive time step procedure for a parabolic problem with blow-up

Abstract In this paper we introduce and analyze a fully discrete approximation for a parabolic problem with a nonlinear boundary condition which implies that the solutions blow up in finite time. We use standard linear elements with mass lumping for the space variable. For the time discretization we write the problem in an equivalent form which is obtained by introducing an appropriate time re-scaling and then, we use explicit Runge-Kutta methods for this equivalent problem. In order to motivate our procedure we present it first in the case of a simple ordinary differential equation and show how the blow up time is approximated in this case. We obtain necessary and sufficient conditions for the blow-up of the numerical solution and prove that the numerical blow-up time converges to the continuous one. We also study, for the explicit Euler approximation, the localization of blow-up points for the numerical scheme.

Error Estimates for

We prove optimal order error estimates in the H1 norm for the

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Isoparametric Elements Satisfying a Weak Angle Condition

isoparametric interpolation on convex quadrilateral elements under rather weak hypotheses, improving previously known results. Choose one diagonal and divide the element into two triangles. We show that, if the chosen diagonal is the longest one, then the constant in the error estimate depends only on the maximum angle of the two triangles. Otherwise, the constant depends on that maximum angle and on the ratio between the two diagonals. In particular, we obtain the optimal order error estimate under the maximum angle condition as in the case of triangular elements. Consequently, the error estimate is uniformly valid for a rather general class of degenerate quadrilaterals.

Anisotropic error estimates for an interpolant defined via moments

SummaryAn interpolant defined via moments is investigated for triangles, quadrilaterals, tetrahedra, and hexahedra and arbitrarily high polynomial degree. The elements are allowed to have diameters with different asymptotic behavior in different space directions. Anisotropic interpolation error estimates are proved.

Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra

We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three-dimensional maximum angle condition and the regular vertex property, for tetrahedra. Our techniques are different from those used in previous papers on the subject, and the results obtained are more general in several aspects. First, intermediate regularity is allowed; that is, for the Raviart-Thomas interpolation of degree k > 0, we prove error estimates of order j + 1 when the vector field being approximated has components in W j+1,p , for triangles or tetrahedra, where 0 < j < k and 1 < p < ∞. These results are new even in the two-dimensional case. Indeed, the estimate was known only in the case j = k. On the other hand, in the three-dimensional case, results under the maximum angle condition were known only for k = 0.

The Full Strategy Minority Game

The Full Strategy Minority Game (FSMG) is an instance of the Minority Game (MG) which includes a single copy of every potential agent. In this work, we explicitly solve the FSMG thanks to certain symmetries of this game. Furthermore, by considering the MG as a statistical sample of the FSMG, we compute approximated values of the key variable σ2/N in the symmetric phase for different versions of the MG. As another application we prove that our results can be easily modified in order to handle certain kinds of initial biased strategy scores, in particular when the bias is introduced at the agents’ level. We also show that the FSMG verifies a strict period two dynamics (i.e., period two dynamics satisfied with probability 1) giving, to the best of our knowledge, the first example of an instance of the MG for which this feature can be analytically proved. Thanks to this property, it is possible to compute in a simple way the probability that a general instance of the MG breaks the period two dynamics for the first time in a given simulation.

Finite element approximations in a non-Lipschitz domain: Part II

Fil: Acosta, Gabriel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matematica; Argentina

The minimal angle condition for quadrilateral finite elements of arbitrary degree

We study W1,p Lagrange interpolation error estimates for general quadrilateralQk finite elements with k2. For the most standard case of p=2 it turns out that the constant C involved in the error estimate can be bounded in terms of the minimal interior angle of the quadrilateral. Moreover, the same holds for any p in the range 1p<3. On the other hand, for 3p we show that C also depends on the maximal interior angle. We provide some counterexamples showing that our results are sharp.