Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827

Computational modelling of iron-ore mineralisation with stratigraphic permeability anisotropy

This study develops a computational framework to model fluid transport in sedimentary basins, targeting iron ore deposit formation. It offers a simplified flow model, accounting for geological features and permeability anisotropy as driving factors. A new finite element method lessens computational effort, facilitating robust predictions and cost-effective exploration. This methodology, applicable to other mineral commodities, enhances understanding of genetic models, supporting the search for new mineral deposits amid the global energy transition

Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors

This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor $L^2$ approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering

A stabilised finite element method for the convection-diffusion-reaction equation in mixed form

This paper is devoted to the approximation of the convection-diffusion-reaction equation using a mixed, first-order, formulation. We propose, and analyse, a stabilised finite element method that allows equal order interpolations for the primal and dual variables. This formulation, reminiscent of the Galerkin least-squares method, is proven stable and convergent. In addition, a numerical assessment of the numerical performance of different stabilised finite element methods for the mixed formulation is carried out, and the different methods are compared in terms of accuracy, stability, and sharpness of the layers for two different classical test problems

Augmenting Numerical Stability of the Galerkin Finite Element Formulation for Electromagnetic Flowmeter Analysis

The magnetic flow meter is one of the best possible choice for the measurement of flow rate of liquid metals in fast breeder reactors. Due to the associated complexities in the measuring environment, theoretical evaluation of their sensitivity is always preferred. In order to consider the 3D nature of the problem and the general flow patterns, numerical field computational approach is inevitable. When classical Galerkin's finite element formulation is employed for the solution, it is known to introduce numerical oscillations at high flow rates. The magnetic field produced by the flow induced currents circulate within the fluid and forms the source of this numerical problem. To overcome this, modified methods like stream-line upwind Petrov-Galerkin schemes are generally suggested in the allied areas like fluid dynamics, in which a similar dominance of advective (curl or circulation) component occurs over diffusion (divergence) component. After a careful analysis of the numerical instability through a reduced one dimensional problem, an elegant stable approach is devised. In this scheme, a pole-zero cancellation approach is adopted. The proposed scheme is shown to be absolutely stable. However, at lower flow rates numerical results exhibits small oscillation, which can be controlled by reducing the element size. The source of stability at higher flow rates, as well as, oscillations at lower flow rates are analysed using analytical solution of the associated difference equation. Finally the proposed approach is applied to the original flow meter problem and the solution is shown to be stable.Comment: IET Science, Measurement & Technology, 201