Approximation of linear functionals using an hp-adaptive discontinuous Galerkin finite element method

We consider the problem of computing a linear functional of the solution of an elliptic partial differential equation to within a given tolerance. We drive an a posteriori error bound for the linear functional and use this as the basis of an hp-adaptive discontinuous Galerkin finite element algorithm to deliver the functional to within a prescribed error tolerance

Application of hpDGFEM to mechanisms at channel microband electrodes

We extend our earlier work (Harriman et al., Oxford University Computing Laboratory Technical Report NA04/19) on hp-DGFEM for disc electrodes to the case of reaction mechanisms to the increasingly popular channel microband electrode configuration. We present results for the simple E reaction mechanism (convection-diffusion equation), for the ECE and EC2E reaction mechanisms (linear and nonlinear systems of reaction-convection- diffusion equations, respectively) and for the DISP1 and DISP2 reaction mechanisms (linear and nonlinear coupled systems of reaction-convection-diffusion equations, respectively). In all cases we demonstrate excellent agreement with previous results using relatively coarse meshes and without the need for streamline-diffusion stabilisation, even at high flow rates

Adaptive Finite Element Simulation of Steady State Currents at Microdisc Electrodes to a Guaranteed Accuracy

We consider the general problem of numerical simulation of the currents at microelectrodes using an adaptive finite element approach. Microelectrodes typically consist of an electrode embedded (or recessed) in an insulating material. For all such electrodes, numerical simulation is made difficult by the presence of a boundary singularity at the electrode edge (where the electrode meets the insulator), manifested by the large increase in the current density at this point, often referred to as the "edge-effect". Our approach to overcoming this problem involves the derivation of an a posteriori bound on the error in the numerical approximation for the current which can be used to drive an adaptive mesh-generation algorithm. This allows us to calculate the current to within a prescribed tolerance. Here we demonstrate the power of the method for a simple model problem -- an E reaction mechanism at a microdisc electrode -- for which the analytical solution is known, then we extend the work to the case of a (pseudo) first order EC' reaction mechanism at both an inlaid and a recessed disc

Adaptive Finite Element Simulation of Currents at Microelectrodes to a Guaranteed Accuracy. Application to Channel Microband Electrodes.

We extend our earlier work (see K. Harriman et al., Technical Report NA99/19) on adaptive finite element methods for disc electrodes to the case of reaction mechanisms to the increasingly popular channel microband electrode configuration. We use the standard Galerkin finite element method for the diffusion-dominated (low-flow) case, and the streamline diffusion finite element method for the convection-dominated (high-flow) case. We first consider the simple E reaction mechanism (convection-diffusion equation) and we demonstrate excellent agreement with previous approximate analytical results across the range of parameters of interest, on comparatively coarse meshes. We then consider ECE and EC2E reaction mechanisms (linear and nonlinear systems of reaction-convection-diffusion equations, respectively); again we are able to demonstrate excellent agreement with previous results.\ud \ud The authors are pleased to acknowledge the financial support of the following organisations: a research studentship for KH; a Career Development Fellowship from the Medical Research Council for DJG, which has allowed them to undertake this research

Measurements of strongly-anisotropic g-factors for spins in single quantum states

We have measured the full angular dependence, as a function of the direction of magnetic field, for the Zeeman splitting of individual energy states in copper nanoparticles. The g-factors for spin splitting are highly anisotropic, with angular variations as large as a factor of five. The angular dependence fits well to ellipsoids. Both the principal-axis directions and g-factor magnitudes vary between different energy levels within one nanoparticle. The variations agree quantitatively with random-matrix theory predictions which incorporate spin-orbit coupling.Comment: 4 pages, 3 figures, 2 in colo

The importance of adjoint consistency in the approximation of linear functionals using the discontinuous Galerkin finite element method

We describe how a discontinuous Galerkin finite element method with interior penalty can be used to compute the solution to an elliptic partial differential equation and a linear functional of this solution can be evaluated. We show that, in order to have an adjoint consistent method and thus obtain optimal rates of convergence of the functional, a symmetric interior penalty Galerkin method must be used and, when the functional depends on the derivative of the solution of the PDE, an equivalent formulation of the functional must be used

Finite element solution of a membrane covered electrode problem

Membrane covered oxygen sensors, or Clark electrodes, are used for monitoring the concentration of oxygen in blood. The operation of such sensors is governed by the diffusion equation with different diffusion coefficients in different sub-domains. The form of the boundary conditions and the material interface conditions means that the derivative of the solution has discontinuities which restrict the convergence of standard numerical methods on regular meshes. We describe and computationally compare adaptive finite element methods based on continuous and discontinuous basis functions to overcome this problem

Mechanical Response of He- Implanted Amorphous SiOC/ Crystalline Fe Nanolaminates

This study investigates the microstructural evolution and mechanical response of sputter-deposited amorphous silicon oxycarbide (SiOC)/crystalline Fe nanolaminates, a single layer SiOC film, and a single layer Fe film subjected to ion implantation at room temperature to obtain a maximum He concentration of 5 at. %. X-ray diffraction and transmission electron microscopy indicated no evidence of implantation-induced phase transformation or layer breakdown in the nanolaminates. Implantation resulted in the formation of He bubbles and an increase in the average size of the Fe grains in the individual Fe layers of the nanolaminates and the single layer Fe film, but the bubble density and grain size were found to be smaller in the former. By reducing the thicknesses of individual layers in the nanolaminates, bubble density and grain size were further decreased. No He bubbles were observed in the SiOC layers of the nanolaminates and the single layer SiOC film. Nanoindentation and scanning probe microscopy revealed an increase in the hardness of both single layer SiOC and Fe films after implantation. For the nanolaminates, changes in hardness were found to depend on the thicknesses of the individual layers, where reducing the layer thickness to 14 nm resulted in mitigation of implantation-induced hardening