We consider H(curl, Ω)-elliptic variational problems on bounded Lipschitz polyhedra and their finite element Galerkin discretization by means of lowest order edge elements. We assume that the underlying tetrahedral mesh has been created by successive local mesh refinement, either by local uniform refinement with hanging nodes or bisection refinement. In this setting we develop a convergence the...

In this paper, we study the problem of computing effective diffusivity for particles moving in chaotic flows. Instead solving a convection-diffusion type cell Eulerian formulation (arising from homogenization theory parabolic equations), compute motion Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). A robust numerical integrator based on splitting method wa...

We consider a finite difference semi-discrete scheme for the approximation of the boundary controls of a 1-D equation modelling the transversal vibrations of a hinged beam. It is known that, due to the high frequency numerical spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural setting. Consequently, the con...

We generalize Brondsted's results in [2] and [3] in order to obtain uniform space versions of Caristi's fixed point theorem, Ekeland's variational principle and the drop theorem. Moreover, it is applied to weak convergence of random iterations.

Noam Nisan constructed pseudo random number generators which convert O(S log R) truly random bits to R bits that appear random to any algorithm that runs in SPACE(S). D Sivakumar, demonstrated that a large class of probabilistic algorithms can be derandomized using Nisan’s construction. This class of algorithms is characterized by the fact that each probabilistic algorithm can thought of as a s...

Let I ⊂ P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M, μ), we obtain a statistical version of the Egorov theorem (when μ(X) < ∞). We show that, in its assertion, equi-statistical convergence...

The convergence in meaaa of a veighted sum k a.k(Xk EXk) of random elements in a separable Banach space is studied under a new hypothesis which relates the random elements with their respective weights in the sum: the {a.. }-compactly uniform integrability of {X. }. This condition, which is implied by the tightness of {X,,} and the {a,,k }-uniform integrability of {[IX,, II}, is weaker than the...

Smale’s 17th Problem asks “Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average [for a suitable probability measure on the space of inputs], in polynomial time with a uniform algorithm?” We present a uniform probabilistic algorithm for this problem and prove that its complexity is polynomial. We thus obtain a partial positive solution to Smale 17th P...

In this paper we study the convergence of a centred finite difference scheme on a non-uniform mesh for a 1D elliptic problem subject to general boundary conditions. On a non-uniform mesh, the scheme is, in general, only first-order consistent. Nevertheless, we prove for s ∈ (1/2, 2] order O(hs)-convergence of solution and gradient if the exact solution is in the Sobolev space H1+s(0, L), i.e. t...