This paper develops a multilevel least-squares approach for the numerical solution of the complex scalar exterior Helmholtz equation. This second-order equation is first recast into an equivalent first-order system by introducing several “field” variables. A combination of scaled L and H norms is then applied to the residual of this system to create a least-squares functional. It is shown that,...

We define the concept of polynomial uniform convergence of relative frequencies to probabilities in the distribution-dependent context. Let Xn = {O, l}n, let Pn be a probability distribution on Xn and let Fn C 2X ,. be a family of events. The family {(Xn, Pn, Fn)}n~l has the property of polynomial uniform convergence if the probability that the maximum difference (over Fn) between the relative ...

Let (X, |·|) be an infinite dimensional Banach space and L(X) be the space of bounded linear operators from X into X. Suppose that r > 0 is a given real number. C ([−r, 0] , X) denotes the space of continuous functions from [−r, 0] to X with the uniform convergence topology and we will use simply CX for C ([−r, 0] , X). For u ∈ C ([−r, b] , X), b > 0 and t ∈ [0, b], let ut denote the element of...

We propose an abstract two-level algorithm with convergence independent of the coarse-space size. The abstract algorithm is applied to problems of solids discretized on unstructured meshes. With no regularity assumptions we prove uniform convergence with respect to coarse-space size, domain, essential boundary conditions and jumps in Young modulus. Numerical experiments connrm the theory and sh...

Let X be a locally connected, b-compact metric space and E a closed subset of X. Let G be the space of all continuous real-valued functions defined on some closed subsets of E. We prove the equivalence of the τ aw and τ K topologies on G, where τ aw is the so called Attouch-Wets topology, defined in terms of uniform convergence of distance functionals, and τ K is the topology of Kuratowski conv...

Let L be a real (Hausdorff) topological vector space. The space %[L] of nonempty compact subsets of L forms a (Hausdorff) topological semivector space with singleton origin when 3C[L] is given the uniform (equivalently, the finite) hyperspace topology determined by L. Then %[L] is locally compact iff L is so. Furthermore, 90.[Z,], the set of nonempty compact convex subsets of L, is the largest ...

in a Banach space E with a strongly positive operator A and with an arbitrary positive parameter ε. We establish the well-posedness in difference analogue of Hölder space of the high order uniform difference scheme for (0.1). Moreover, in applications, the convergence estimates for the solutions of uniform difference schemes of the multidimensional parabolic differential equations with an arbit...

In this work, the bilinear nite element method on a Shishkin mesh for convection-diiusion problems is analyzed in the two-dimensional setting. A su-perconvergence rate O(N ?2 ln 2 N + N ?1:5 ln N) in a discrete-weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter. Numerical tests in...