In this paper we present a new a posteriori error estimate for the boundary element method applied to an integral equation of the first kind. The estimate is local and sharp for quasi-uniform meshes and so improves earlier work of ours. The mesh-dependence of the constants is analyzed and shown to be weaker than expected from our previous work. Besides the Galerkin boundary element method, the ...

In this note, we make a few comments concerning the paper of Hughes and Akin [4]. Our primary goal is to demonstrate that the rate of convergence of numerical solutions of the finite element method with singular basis functions depends upon the location of additional collocation points associated with the singular elements.

In this article, we study superconvergence properties of immersed finite element methods for the one dimensional elliptic interface problem. Due to low global regularity of the solution, classical superconvergence phenomenon for finite element methods disappears unless the discontinuity of the coefficient is resolved by partition. We show that immersed finite element solutions inherit all desir...

Let the parabolic problem cix, t, u)ut = aix, t, u)uxx + bix, t, u, ux), 0 < x < 1, 0 < / á T, uix, 0) = fix), w(0, t) = gli), ií(1, t) = giit), be solved approximately by the continuous-time collocation process based on having the differential equation satisfied at Gaussian points £,,i and £;,2 in subintervals (x,-_i, x¡) for a function l/:[0, T] —» 3C3, the class of Hermite piecewise-cubic po...

A quick and easy computation involving magnetic rods is presented. Rods are replaced by different kinds of distributions (magnetization, magnetic charges) calculated by collocation. These distributions can be punctual or surfacic (disc) in the rod according to the density of the mesh used. Then, three-dimensional geometries can be quickly and easily modeled instead of using uneasy finite-elemen...

In this work, we analyse new robust spline approximation methods for mth order boundary value problems described by nonlinear ordinary diierential and integro-diierential equations with m linear boundary conditions. Our main aim is to introduce a cost-eeective alternative to the highly successful orthogonal col-location method, and to prove stability and convergence properties similar to the or...

The built up layer thickness in secondary deformation zone is one of the important parameters in metal cutting process. The built up layer (BUL) is formed in second deformation zone near the tool-chip interface in the back of the chip. This parameter influences the tool life and machined surface quality. This BUL should not be confused with the built up edge (BUE). The deformation of the BUL in...

The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown deterministic component and a known zero-mean stochastic component. The extra freedom provided by t...