A. J. Meir

Bio: A. J. Meir is an academic researcher from Auburn University. The author has contributed to research in topics: Finite element method & Boundary value problem. The author has an hindex of 15, co-authored 26 publications receiving 708 citations. Previous affiliations of A. J. Meir include University of Alabama.

On the existence, uniqueness, and finite element approximation of solutions of the equations of stationary, incompressible magnetohydrodynamics

Abstract: The authors consider the equations of stationary, incompressible magneto-hydrodynamics posed in a bounded domain in three dimensions and treat the full, coupled system of equations with inhomogeneous boundary conditions. Under certain conditions on the data, they show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. They discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution.

Analysis and Numerical Approximation of a Stationary MHD Flow Problem with Nonideal Boundary

Abstract: We are concerned with the steady flow of a conducting fluid, confined to a bounded region of space and driven by a combination of body forces, externally generated magnetic fields, and currents entering and leaving the fluid through electrodes attached to the surface. The flow is governed by the Navier--Stokes equations (in the fluid region) and Maxwell's equations (in all of space), coupled via Ohm's law and the Lorentz force. By means of the Biot--Savart law, we reduce the problem to a system of integro-differential equations in the fluid region, derive a mixed variational formulation, and prove its well-posedness under a small-data assumption. We then study the finite-element approximation of solutions (in the case of unique solvability) and establish optimal-order error estimates. Finally, an implementation of the method is described and illustrated with the results of some numerical experiments.

Dynamic diffusion model for tracing the real-time potential response of polymeric membrane ion-selective electrodes.

Abstract: A numerical solution for the prediction of the time-dependent potential response of a polymeric-based ion-selective electrode (ISE) is presented. The model addresses short- and middle-term potential drifts that are dependent on changes in concentration gradients in the aqueous sample and organic membrane phase. This work has important implications for the understanding of the real-time response behavior of potentiometric sensors with low detection limits and with nonclassical super-Nernstian response slopes. As a model system, the initial exposure of membranes containing the well-examined silver ionophore O,O‘ ‘-bis[2-(methylthio)ethyl]-tert-butylcalix[4]arene was monitored, and the large observed potential drifts were compared to theoretical predictions. The model is based on an approximate solution of the diffusion equation for both aqueous and organic diffusion layers using a numerical scheme (finite difference in time and finite elements in space). The model may be evaluated on the basis of experiment...

A two-level discretization method for the stationary mhd equations

Abstract: We describe and analyze a two-level finite-element method for discretizing the equations of stationary, viscous, incompressible magnetohydrodynamics (or MHD) These equations, which model the flow of electrically conducting fluids in the presence of electromagnetic fields, arise in plasma physics and liquid-metal technology as well as in geophysics and astronomy We treat the equations under physically realistic ("nonideal") boundary conditions that account for the electromagnetic interaction of the fluid with the surrounding media The suggested algorithm involves solving a small, nonlinear problem on a coarse mesh and then one large, linear problem on a fine mesh We prove well-posedness of the algorithm and optimal error estimates under a small-data assumption

A Two-Grid Method of a Mixed Stokes-Darcy Model for Coupling Fluid Flow with Porous Media Flow

Abstract: We study numerical methods for solving a coupled Stokes-Darcy problem in porous media flow applications. A two-grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions. Error estimates are derived for the proposed method. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two-grid approach for solving multimodeling problems. Potential extensions and future directions are discussed.

Solid contact potentiometric sensors for trace level measurements.

Abstract: A simple procedure for the development of a range of polymeric ion-selective electrodes (ISEs) with low detection limits is presented. The electrodes were prepared by using a plasticizer-free methyl methacrylate−decyl methacrylate copolymer as membrane matrix and poly(3-octylthiophene) as intermediate layer deposited by solvent casting on gold sputtered copper electrodes as a solid inner contact. Five different electrodes were developed for Ag+, Pb2+, Ca2+, K+, and I-, with detection limits mostly in the nanomolar range. In this work, the lowest detection limits reported thus far with solid contact ISEs for the detection of silver (2.0 × 10-9 M), potassium (10-7 M), and iodide (10-8 M) are presented. The developed electrodes exhibited a good response time and excellent reproducibility.

Mixed finite element methods for stationary incompressible magneto–hydrodynamics

Abstract: A new mixed variational formulation of the equations of stationary incompressible magneto–hydrodynamics is introduced and analyzed. The formulation is based on curl-conforming Sobolev spaces for the magnetic variables and is shown to be well-posed in (possibly non-convex) Lipschitz polyhedra. A finite element approximation is proposed where the hydrodynamic unknowns are discretized by standard inf-sup stable velocity-pressure space pairs and the magnetic ones by a mixed approach using Nedelec’s elements of the first kind. An error analysis is carried out that shows that the proposed finite element approximation leads to quasi-optimal error bounds in the mesh-size.

Comput. Methods Appl. Mech. Engrg.

Abstract: There exist two types of commonly studied stochastic elliptic models in literature: (I) � r � (a(x,x)r u(x,x)) = f(x) and (II) � r �ð aðx; xÞ}ruðx;xÞÞ ¼ f ðxÞ, where x indicates randomness, } the Wick product, and a(x,x) is a positive random process. Model (I) is widely used in engineering and physical applications while model (II) is usually studied from the mathematical point of view. The difference between the above two stochastic elliptic models has not been fully clarified. In this work, we discuss the difference between models (I) and (II) when a(x,x) is a log-normal random process. We show that the difference between models (I) and (II) is mainly characterized by a scaling factor, which is an exponential function of the degree of perturbation of a(x,x). We then construct a new stochastic elliptic model (III): � r �ð ða