Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form $D = \{(x, z)\in \mathbb{R}^{n+1} : x\in \mathbb{R}^n, z > f(x)\}$ where $f : \mathbb{R}^n \to\mathbb{R}$ is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and problems in potential theory, have been reformulated as second kind integral equations $u+Ku = v$ in the space $BC$ of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator $A = I + K$ under consideration, with an emphasis on the function space setting $BC$. Firstly, under which conditions is $A$ a Fredholm operator, and, secondly, when is the finite section method applicable to $A$?

Negative correlations between control performance and verbalizable knowledge: indicators for implicit learning in process control tasks?

Negative correlations between task performance in dynamic control tasks and verbalizable knowledge, as assessed by a post-task questionnaire, have been interpreted as dissociations that indicate two antagonistic modes of learning, one being “explicit”, the other “implicit”. This paper views the control tasks as finite-state automata and offers an alternative interpretation of these negative correlations. It is argued that “good controllers” observe fewer different state transitions and, consequently, can answer fewer post-task questions about system transitions than can “bad controllers”. Two experiments demonstrate the validity of the argument by showing the predicted negative relationship between control performance and the number of explored state transitions, and the predicted positive relationship between the number of explored state transitions and questionnaire scores. However, the experiments also elucidate important boundary conditions for the critical effects. We discuss the implications of these findings, and of other problems arising from the process control paradigm, for conclusions about implicit versus explicit learning processes.

Finite element approximation of a sixth order nonlinear degenerate parabolic equation

We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation ut = ?.( b(u)? 2u), where generically b(u) := |u|? for any given ? ? (0,?). In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d ? 3. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments in one and two space dimensions are presented.

Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold. (c) 2005 IMACS. Published by Elsevier B.V. All rights reserved.

An adaptive finite element water column model using the Mellor-Yamada level 2.5 turbulence closure scheme

A one-dimensional water column model using the Mellor and Yamada level 2.5 parameterization of vertical turbulent fluxes is presented. The model equations are discretized with a mixed finite element scheme. Details of the finite element discrete equations are given and adaptive mesh refinement strategies are presented. The refinement criterion is an "a posteriori" error estimator based on stratification, shear and distance to surface. The model performances are assessed by studying the stress driven penetration of a turbulent layer into a stratified fluid. This example illustrates the ability of the presented model to follow some internal structures of the flow and paves the way for truly generalized vertical coordinates. (c) 2005 Elsevier Ltd. All rights reserved.