AJ Meir - Academia.edu (original) (raw)

Papers by AJ Meir

Research paper thumbnail of An Inverse Diffusion Coefficient Problem for a Parabolic Equation with Integral Constraint

We consider a problem of recovering the time-dependent diffusion coefficient in a parabolic syste... more We consider a problem of recovering the time-dependent diffusion coefficient in a parabolic system. To ensure uniqueness the system is constrained by the integral of the solution at all times. This problem has applications in geology where the parabolic equation models the accumulation and diffusion of argon in micas. Argon is generated by the decay of potassium and the diffusion is thermally activated. We introduce a time discretization, on which we base an application of Rothe’s method to prove existence of solutions. The numerical scheme corresponding to the semi-discretization exhibits convergence that is consistent with that in Euler’s method.

Research paper thumbnail of On Some Problems in Poroelasticity

Poromechanics is the science of energy, motion, and forces, and their effect on porous material a... more Poromechanics is the science of energy, motion, and forces, and their effect on porous material and in particular the swelling and shrinking of fluid-saturated porous media. Modeling and predicting the mechanical behavior of fluid-infiltrated porous media is significant since many natural substances, for example, rocks, soils, clays, shales, biological tissues, and bones, as well as man-made materials, such as, foams, gels, concrete, water-solute drug carriers, and ceramics are all elastic porous materials (hence poroelastic). In this talk I will describe some problems in poroelasticity and their mathematical analysis. I will also describe finite element based numerical methods for approximating solutions of model problems in poroelasticity, and the available a-priori error estimates.

Research paper thumbnail of Numerical approximation of solutions of the equations of quasistatic electroporoelasticity

Numerical Methods for Partial Differential Equations, 2021

Research paper thumbnail of A locking free numerical approximation for quasilinear poroelasticity problems

Computers & Mathematics with Applications, 2020

This paper is devoted to a new formulation which couples the weak Galerkin method and the finite ... more This paper is devoted to a new formulation which couples the weak Galerkin method and the finite element method for approximating solutions of the equations of quasistatic poroelasticity which model flow through elastic porous media. It is assumed that the permeability of the elastic matrix depends nonlinearly on the dilatation of the porous medium. The steady-state version of the system is recast in terms of displacement, pressure, and volumetric stress, and the well-posedness of both the continuous and discrete three-field formulations is proved. Error estimates for the proposed numerical method are obtained. These show that the method is locking free. Numerical experiments presented further demonstrate the accuracy and the locking free characteristic of the proposed numerical method.

Research paper thumbnail of An integral constrained parabolic problem with applications in thermochronology

Computers & Mathematics with Applications, 2016

We investigate an inverse source problem with an integral constraint for a parabolic equation. Th... more We investigate an inverse source problem with an integral constraint for a parabolic equation. The constraint is motivated by an application in thermochronology, a branch of geology. The existence and uniqueness of weak solutions are established by means of the Rothe method and an energy method, respectively. The elliptic problem resulting from the time discretization is solved by homogenizing the integral constraint. The implicit scheme used in the proof of existence lends itself readily for numerical studies and we present the results of numerical experiments. We also report on the errors and convergence rates.

Research paper thumbnail of An elliptic problem with integral constraints with application to large-scale geophysical flows

Computational Geosciences

We present a weak formulation of a non-standard elliptic equation whose boundary values are deter... more We present a weak formulation of a non-standard elliptic equation whose boundary values are determined in part by integral relations. Existence and uniqueness of its solution are proved, and a finite element discretization is described, analyzed, and implemented on a test problem. The equation is a generalization of one that is solved during integration of the three-dimensional Quasigeostrophic equations, which model large-scale rotating stratified flows, where the integral constraints represent conservation of physical properties.

Research paper thumbnail of Analysis and Finite Element Simulation of MHD Flows, with an Application to Seawater Drag REDUCTION1

Much research effort has recently been devoted to the electromagnetic control of saltwater flows,... more Much research effort has recently been devoted to the electromagnetic control of saltwater flows, exploiting the macroscopic interaction of saltwater with electric currents and magnetic fields. This interaction is governed by the equations of viscous incompressible MHD, essentially, the Navier-Stokes equations coupled to Maxwell's equations. A major problem in the analysis and numerical solution of these equations is the fact that while the Navier-Stokes equations are posed in the fluid domain, Maxwell's equations are generally posed on all of space. Consequently, electric and magnetic fields do not satisfy standard boundary conditions, but jump or continuity relations on the surface of the fluid domain (and other interfaces). Frequently the resulting difficulties are circumvented by prescribing more or less artificial boundary conditions. In this paper we present a novel formulation of the MHD equations that avoids some inherent difficulties of more traditional approaches...

Research paper thumbnail of Approximation of an optimal convective heat transfer coefficient

Optimal Control Applications and Methods, 2006

ABSTRACT The problem studied in this paper is that of a heat source (electronic chip) placed on t... more ABSTRACT The problem studied in this paper is that of a heat source (electronic chip) placed on the top surface of a flat thermal spreader which is cooled by convection on the opposite surface. An optimal convection heat transfer coefficient yielding maximal heat removal from the chip is found using an optimal control technique. We control the solution of the heat equation with the convective boundary condition, taking the heat transfer coefficient as the control. We show the existence and uniqueness of an optimal control. A conjugate gradient method is used to solve the optimal control problem. The results show that the temperature distributions corresponding to the controlled solution are lower and display a flatter profile than those corresponding to the uncontrolled solution. This study can provide guidance in designing micro heat pipe heat sinks, which have emerged as an effective technique for cooling electronic components. Copyright © 2006 John Wiley & Sons, Ltd.

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

Mathematics of Computation, 1991

We consider the equations of stationary, incompressible magnetohydrodynamics posed in a bounded d... more We consider the equations of stationary, incompressible magnetohydrodynamics 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, we show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. We discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution.

Research paper thumbnail of Experimental Measurements of Velocity, Potential, and Temperature Distributions in Liquid Aluminum During Electromagnetic Stirring

Journal of Applied Mechanics, 2003

An experimental technique has been developed to measure both axial and transverse velocities and ... more An experimental technique has been developed to measure both axial and transverse velocities and temperature distribution in molten aluminum. Couette flow of liquid aluminum, lead, tin, and low melting alloy in cylindrical container was chosen for calibration of the experimental technique and the magnetic probe. Velocity and temperature profiles for liquid aluminum rotating in cylindrical container at different angular velocities are obtained for two different values of the depth. We determined that the velocity values increase with magnetic induction.

Research paper thumbnail of Dynamic Diffusion Model for Tracing the Real-Time Potential Response of Polymeric Membrane Ion-Selective Electrodes

Analytical Chemistry, 2004

A weak form of the equation (obtained by multiplying the equation by a suitable test function υ a... more A weak form of the equation (obtained by multiplying the equation by a suitable test function υ and integrating over !) is: find c such that c, t " # Dc, x ", x

Research paper thumbnail of Normal Pulse Voltammetry as Improved Quantitative Detection Mode for Amperometric Solvent Polymeric Membrane Ion Sensors

Electroanalysis, 2000

A normal pulse voltammetric detection mode for amperometric solvent polymeric membrane ion sensor... more A normal pulse voltammetric detection mode for amperometric solvent polymeric membrane ion sensors is described. These sensors function on the basis of ion transfer voltammetry into an organic membrane phase of high viscosity. To avoid sensor drift, it is required that sample ions extracted within a measurement period are quantitatively stripped off the sensing membrane before the next measurement step. The time required for complete back extraction of previously extracted ions must be substantially longer than for the uptake process. Indeed, more than 40% of extracted ions are predicted to remain in the membrane phase if the stripping time equals the uptake time. This suggests that cyclic voltammetry is generally an inadequate method for a reliable application/characterization of these sensors. The pulsed method imposes discrete potential pulses onto the membrane that are incrementally changing with time to cover the total desired potential range. Between each uptake pulse a suf®ciently long stripping pulse around 0 V is applied. Optimization of uptake and stripping times are performed, and comparative data with cyclic voltammetry are shown. Normal pulse voltammetric detection scans show strictly the current response for the ion uptake process, and are free of superimposed stripping waves. This characteristic aids in elucidating the nature of each observed wave and can therefore also be used for qualitative purposes. The scans also show higher sensitivity than in classical cyclic voltammetry. Experiments are here limited to ionophore-free membranes as model systems.

Research paper thumbnail of Velocity, Potential, and Temperature Distributions in Molten Metals During Electromagnetic Stirring: Part I — Experimental Measurements

Rheology and Fluid Mechanics of Nonlinear Materials, 1999

An experimental technique has been developed to measure the local velocity in molten metals. Coue... more An experimental technique has been developed to measure the local velocity in molten metals. Couette flow of liquid aluminum, lead, tin and low melting alloy in cylindrical container was chosen for calibration of the experimental technique and the magnetic probe. Velocity and temperature profiles for liquid aluminum rotating in cylindrical container at different angular velocities are obtained for two different values of the depth. We determined that the velocity values increase with magnetic induction, and the relationship between the normalized azimuthal velocity and the magnetic induction can be expressed by quadratic function.

Research paper thumbnail of Towards understanding crime dynamics in a heterogeneous environment: A mathematical approach

Journal of Interdisciplinary Mathematics, 2021

Crime data provides information on the nature and location of the crime but, in general, does not... more Crime data provides information on the nature and location of the crime but, in general, does not include information on the number of criminals operating in a region. By contrast, many approaches to crime reduction necessarily involve working with criminals or individuals at risk of engaging in criminal activity and so the dynamics of the criminal population is important. With this in mind, we develop a mechanistic, mathematical model which combines the number of crimes and number of criminals to create a dynamical system. Analysis of the model highlights a threshold for criminal efficiency, below which criminal numbers will settle to an equilibrium level that can be exploited to reduce crime through prevention. This efficiency measure arises from the initiation of new criminals in response to observation of criminal activity; other initiation routes-via opportunism or peer pressure-do not exhibit such thresholds although they do impact on the level of criminal activity observed. We used data from Cape Town, South Africa, to obtain parameter estimates and predicted that the number of criminals in the region is tending towards an equilibrium point but in a heterogeneous manner-a drop in the number of criminals from low crime neighbourhoods is being offset by an increase from high crime neighbourhoods.

Research paper thumbnail of Implementation Of An Efficient Parallel 3-D Finite Element Solver For The Equations of Viscous, Resistive Magnetohydrodynamics Using A Velocity-Current Formulation

Bulletin of the American Physical Society, 2016

Research paper thumbnail of On the equations of poroelasticity and the numerical approximation of their solutions

Poromechanics is the science of energy, motion, and forces, and their effect on porous material a... more Poromechanics is the science of energy, motion, and forces, and their effect on porous material and in particular, the swelling and shrinking of fluid-saturated porous media. Modeling and predicting the mechanical behavior of fluid-infiltrated porous media is significant since many natural substances, for example, rocks, soils, clays, shales, biological tissues, and bones, as well as man-made materials, such as, foams, gels, concrete, water-solute drug carriers, and ceramics are all elastic porous materials (hence poroelastic). Poromechanics is a complex coupled, multiphysics, multiscale, phenomena, where the swelling and shrinking of an elastic deforming porous medium is coupled to the response of the saturating fluid. Other effects, such as, electromechanical, thermal, and chemical may also be considered. After a brief overview, I will describe some nonlinear problems in poroelasticity and their mathematical analysis. I will also describe finite element based numerical methods tha...

Research paper thumbnail of Velocity-vorticity formulation of three-dimensional, steady, viscous, incompressible flows

Research paper thumbnail of On a parallel, 3-dimensional, finite element solver for viscous, resistive, stationary magnetohydrodynamics equations: Velocity–current formulation

Applied Numerical Mathematics, 2018

We describe a parallel implementation for the numerical approximation of solutions to the three-d... more We describe a parallel implementation for the numerical approximation of solutions to the three-dimensional viscous, resistive magnetohydrodynamics (MHD) equations using a velocity-current formulation. In comparison to other formulations, the velocity-current formulation presented in this paper is an integro-differential system of equations that incorporates nonideal boundaries and nonlinearities due to induction. The solution to the equations is approximated using a Picard iteration, discretized with the finite element method, and solved iteratively with the Krylov subspace method GMRES. Effective preconditioning strategies are required to numerically solve the resulting equations with Krylov solvers [12]. For GMRES convergence, the system matrix resulting from the discretization of the velocity-current formulation is preconditioned using a simple, blockdiagonal Schur-complement preconditioner based on [14]. The MHD solver is implemented using freely available, well-documented, open-source, libraries deal.II, p4est, Trilinos, and PETSc, capable of scaling to tens of thousands of processors on state-of-the-art HPC architectures.

Research paper thumbnail of Quasilinear poroelasticity: Analysis and hybrid finite element approximation

Numerical Methods for Partial Differential Equations, 2014

We consider a system of partial differential equations which models flows through elastic porous ... more We consider a system of partial differential equations which models flows through elastic porous media. This system consists of an elasticity equation describing the displacement of an elastic porous matrix and a quasilinear elliptic equation describing the pressure of the saturating fluid (flowing through its pores). In this model, the permeability depends nonlinearly on the dilatation (divergence of the displacement) of the medium. We show that the solution has W 2 m regularity. We describe the numerical approximation of solutions using a hybrid finite element-least squares mixed finite element method. Error estimates are obtained through the introduction of an auxiliary linear elasticity equation. Numerical experiments verify the error estimates and validate the proposed poroelasticity model.

Research paper thumbnail of A First Order Scheme for Backward Doubly Stochastic Differential Equations

SIAM/ASA Journal on Uncertainty Quantification, 2016

In this paper, a fully discrete finite difference method for forward backward doubly stochastic d... more In this paper, a fully discrete finite difference method for forward backward doubly stochastic differential equation is studied. A first order numerical algorithm is obtained using the two sided Itô-Taylor expansion. Numerical experiments verify the accuracy and efficiency of the numerical algorithm. A nonlinear tracking problem is simulated using the first order algorithm.

Research paper thumbnail of An Inverse Diffusion Coefficient Problem for a Parabolic Equation with Integral Constraint

We consider a problem of recovering the time-dependent diffusion coefficient in a parabolic syste... more We consider a problem of recovering the time-dependent diffusion coefficient in a parabolic system. To ensure uniqueness the system is constrained by the integral of the solution at all times. This problem has applications in geology where the parabolic equation models the accumulation and diffusion of argon in micas. Argon is generated by the decay of potassium and the diffusion is thermally activated. We introduce a time discretization, on which we base an application of Rothe’s method to prove existence of solutions. The numerical scheme corresponding to the semi-discretization exhibits convergence that is consistent with that in Euler’s method.

Research paper thumbnail of On Some Problems in Poroelasticity

Poromechanics is the science of energy, motion, and forces, and their effect on porous material a... more Poromechanics is the science of energy, motion, and forces, and their effect on porous material and in particular the swelling and shrinking of fluid-saturated porous media. Modeling and predicting the mechanical behavior of fluid-infiltrated porous media is significant since many natural substances, for example, rocks, soils, clays, shales, biological tissues, and bones, as well as man-made materials, such as, foams, gels, concrete, water-solute drug carriers, and ceramics are all elastic porous materials (hence poroelastic). In this talk I will describe some problems in poroelasticity and their mathematical analysis. I will also describe finite element based numerical methods for approximating solutions of model problems in poroelasticity, and the available a-priori error estimates.

Research paper thumbnail of Numerical approximation of solutions of the equations of quasistatic electroporoelasticity

Numerical Methods for Partial Differential Equations, 2021

Research paper thumbnail of A locking free numerical approximation for quasilinear poroelasticity problems

Computers & Mathematics with Applications, 2020

This paper is devoted to a new formulation which couples the weak Galerkin method and the finite ... more This paper is devoted to a new formulation which couples the weak Galerkin method and the finite element method for approximating solutions of the equations of quasistatic poroelasticity which model flow through elastic porous media. It is assumed that the permeability of the elastic matrix depends nonlinearly on the dilatation of the porous medium. The steady-state version of the system is recast in terms of displacement, pressure, and volumetric stress, and the well-posedness of both the continuous and discrete three-field formulations is proved. Error estimates for the proposed numerical method are obtained. These show that the method is locking free. Numerical experiments presented further demonstrate the accuracy and the locking free characteristic of the proposed numerical method.

Research paper thumbnail of An integral constrained parabolic problem with applications in thermochronology

Computers & Mathematics with Applications, 2016

We investigate an inverse source problem with an integral constraint for a parabolic equation. Th... more We investigate an inverse source problem with an integral constraint for a parabolic equation. The constraint is motivated by an application in thermochronology, a branch of geology. The existence and uniqueness of weak solutions are established by means of the Rothe method and an energy method, respectively. The elliptic problem resulting from the time discretization is solved by homogenizing the integral constraint. The implicit scheme used in the proof of existence lends itself readily for numerical studies and we present the results of numerical experiments. We also report on the errors and convergence rates.

Research paper thumbnail of An elliptic problem with integral constraints with application to large-scale geophysical flows

Computational Geosciences

We present a weak formulation of a non-standard elliptic equation whose boundary values are deter... more We present a weak formulation of a non-standard elliptic equation whose boundary values are determined in part by integral relations. Existence and uniqueness of its solution are proved, and a finite element discretization is described, analyzed, and implemented on a test problem. The equation is a generalization of one that is solved during integration of the three-dimensional Quasigeostrophic equations, which model large-scale rotating stratified flows, where the integral constraints represent conservation of physical properties.

Research paper thumbnail of Analysis and Finite Element Simulation of MHD Flows, with an Application to Seawater Drag REDUCTION1

Much research effort has recently been devoted to the electromagnetic control of saltwater flows,... more Much research effort has recently been devoted to the electromagnetic control of saltwater flows, exploiting the macroscopic interaction of saltwater with electric currents and magnetic fields. This interaction is governed by the equations of viscous incompressible MHD, essentially, the Navier-Stokes equations coupled to Maxwell's equations. A major problem in the analysis and numerical solution of these equations is the fact that while the Navier-Stokes equations are posed in the fluid domain, Maxwell's equations are generally posed on all of space. Consequently, electric and magnetic fields do not satisfy standard boundary conditions, but jump or continuity relations on the surface of the fluid domain (and other interfaces). Frequently the resulting difficulties are circumvented by prescribing more or less artificial boundary conditions. In this paper we present a novel formulation of the MHD equations that avoids some inherent difficulties of more traditional approaches...

Research paper thumbnail of Approximation of an optimal convective heat transfer coefficient

Optimal Control Applications and Methods, 2006

ABSTRACT The problem studied in this paper is that of a heat source (electronic chip) placed on t... more ABSTRACT The problem studied in this paper is that of a heat source (electronic chip) placed on the top surface of a flat thermal spreader which is cooled by convection on the opposite surface. An optimal convection heat transfer coefficient yielding maximal heat removal from the chip is found using an optimal control technique. We control the solution of the heat equation with the convective boundary condition, taking the heat transfer coefficient as the control. We show the existence and uniqueness of an optimal control. A conjugate gradient method is used to solve the optimal control problem. The results show that the temperature distributions corresponding to the controlled solution are lower and display a flatter profile than those corresponding to the uncontrolled solution. This study can provide guidance in designing micro heat pipe heat sinks, which have emerged as an effective technique for cooling electronic components. Copyright © 2006 John Wiley & Sons, Ltd.

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

Mathematics of Computation, 1991

We consider the equations of stationary, incompressible magnetohydrodynamics posed in a bounded d... more We consider the equations of stationary, incompressible magnetohydrodynamics 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, we show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. We discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution.

Research paper thumbnail of Experimental Measurements of Velocity, Potential, and Temperature Distributions in Liquid Aluminum During Electromagnetic Stirring

Journal of Applied Mechanics, 2003

An experimental technique has been developed to measure both axial and transverse velocities and ... more An experimental technique has been developed to measure both axial and transverse velocities and temperature distribution in molten aluminum. Couette flow of liquid aluminum, lead, tin, and low melting alloy in cylindrical container was chosen for calibration of the experimental technique and the magnetic probe. Velocity and temperature profiles for liquid aluminum rotating in cylindrical container at different angular velocities are obtained for two different values of the depth. We determined that the velocity values increase with magnetic induction.

Research paper thumbnail of Dynamic Diffusion Model for Tracing the Real-Time Potential Response of Polymeric Membrane Ion-Selective Electrodes

Analytical Chemistry, 2004

A weak form of the equation (obtained by multiplying the equation by a suitable test function υ a... more A weak form of the equation (obtained by multiplying the equation by a suitable test function υ and integrating over !) is: find c such that c, t " # Dc, x ", x

Research paper thumbnail of Normal Pulse Voltammetry as Improved Quantitative Detection Mode for Amperometric Solvent Polymeric Membrane Ion Sensors

Electroanalysis, 2000

A normal pulse voltammetric detection mode for amperometric solvent polymeric membrane ion sensor... more A normal pulse voltammetric detection mode for amperometric solvent polymeric membrane ion sensors is described. These sensors function on the basis of ion transfer voltammetry into an organic membrane phase of high viscosity. To avoid sensor drift, it is required that sample ions extracted within a measurement period are quantitatively stripped off the sensing membrane before the next measurement step. The time required for complete back extraction of previously extracted ions must be substantially longer than for the uptake process. Indeed, more than 40% of extracted ions are predicted to remain in the membrane phase if the stripping time equals the uptake time. This suggests that cyclic voltammetry is generally an inadequate method for a reliable application/characterization of these sensors. The pulsed method imposes discrete potential pulses onto the membrane that are incrementally changing with time to cover the total desired potential range. Between each uptake pulse a suf®ciently long stripping pulse around 0 V is applied. Optimization of uptake and stripping times are performed, and comparative data with cyclic voltammetry are shown. Normal pulse voltammetric detection scans show strictly the current response for the ion uptake process, and are free of superimposed stripping waves. This characteristic aids in elucidating the nature of each observed wave and can therefore also be used for qualitative purposes. The scans also show higher sensitivity than in classical cyclic voltammetry. Experiments are here limited to ionophore-free membranes as model systems.

Research paper thumbnail of Velocity, Potential, and Temperature Distributions in Molten Metals During Electromagnetic Stirring: Part I — Experimental Measurements

Rheology and Fluid Mechanics of Nonlinear Materials, 1999

An experimental technique has been developed to measure the local velocity in molten metals. Coue... more An experimental technique has been developed to measure the local velocity in molten metals. Couette flow of liquid aluminum, lead, tin and low melting alloy in cylindrical container was chosen for calibration of the experimental technique and the magnetic probe. Velocity and temperature profiles for liquid aluminum rotating in cylindrical container at different angular velocities are obtained for two different values of the depth. We determined that the velocity values increase with magnetic induction, and the relationship between the normalized azimuthal velocity and the magnetic induction can be expressed by quadratic function.

Research paper thumbnail of Towards understanding crime dynamics in a heterogeneous environment: A mathematical approach

Journal of Interdisciplinary Mathematics, 2021

Crime data provides information on the nature and location of the crime but, in general, does not... more Crime data provides information on the nature and location of the crime but, in general, does not include information on the number of criminals operating in a region. By contrast, many approaches to crime reduction necessarily involve working with criminals or individuals at risk of engaging in criminal activity and so the dynamics of the criminal population is important. With this in mind, we develop a mechanistic, mathematical model which combines the number of crimes and number of criminals to create a dynamical system. Analysis of the model highlights a threshold for criminal efficiency, below which criminal numbers will settle to an equilibrium level that can be exploited to reduce crime through prevention. This efficiency measure arises from the initiation of new criminals in response to observation of criminal activity; other initiation routes-via opportunism or peer pressure-do not exhibit such thresholds although they do impact on the level of criminal activity observed. We used data from Cape Town, South Africa, to obtain parameter estimates and predicted that the number of criminals in the region is tending towards an equilibrium point but in a heterogeneous manner-a drop in the number of criminals from low crime neighbourhoods is being offset by an increase from high crime neighbourhoods.

Research paper thumbnail of Implementation Of An Efficient Parallel 3-D Finite Element Solver For The Equations of Viscous, Resistive Magnetohydrodynamics Using A Velocity-Current Formulation

Bulletin of the American Physical Society, 2016

Research paper thumbnail of On the equations of poroelasticity and the numerical approximation of their solutions

Poromechanics is the science of energy, motion, and forces, and their effect on porous material a... more Poromechanics is the science of energy, motion, and forces, and their effect on porous material and in particular, the swelling and shrinking of fluid-saturated porous media. Modeling and predicting the mechanical behavior of fluid-infiltrated porous media is significant since many natural substances, for example, rocks, soils, clays, shales, biological tissues, and bones, as well as man-made materials, such as, foams, gels, concrete, water-solute drug carriers, and ceramics are all elastic porous materials (hence poroelastic). Poromechanics is a complex coupled, multiphysics, multiscale, phenomena, where the swelling and shrinking of an elastic deforming porous medium is coupled to the response of the saturating fluid. Other effects, such as, electromechanical, thermal, and chemical may also be considered. After a brief overview, I will describe some nonlinear problems in poroelasticity and their mathematical analysis. I will also describe finite element based numerical methods tha...

Research paper thumbnail of Velocity-vorticity formulation of three-dimensional, steady, viscous, incompressible flows

Research paper thumbnail of On a parallel, 3-dimensional, finite element solver for viscous, resistive, stationary magnetohydrodynamics equations: Velocity–current formulation

Applied Numerical Mathematics, 2018

We describe a parallel implementation for the numerical approximation of solutions to the three-d... more We describe a parallel implementation for the numerical approximation of solutions to the three-dimensional viscous, resistive magnetohydrodynamics (MHD) equations using a velocity-current formulation. In comparison to other formulations, the velocity-current formulation presented in this paper is an integro-differential system of equations that incorporates nonideal boundaries and nonlinearities due to induction. The solution to the equations is approximated using a Picard iteration, discretized with the finite element method, and solved iteratively with the Krylov subspace method GMRES. Effective preconditioning strategies are required to numerically solve the resulting equations with Krylov solvers [12]. For GMRES convergence, the system matrix resulting from the discretization of the velocity-current formulation is preconditioned using a simple, blockdiagonal Schur-complement preconditioner based on [14]. The MHD solver is implemented using freely available, well-documented, open-source, libraries deal.II, p4est, Trilinos, and PETSc, capable of scaling to tens of thousands of processors on state-of-the-art HPC architectures.

Research paper thumbnail of Quasilinear poroelasticity: Analysis and hybrid finite element approximation

Numerical Methods for Partial Differential Equations, 2014

We consider a system of partial differential equations which models flows through elastic porous ... more We consider a system of partial differential equations which models flows through elastic porous media. This system consists of an elasticity equation describing the displacement of an elastic porous matrix and a quasilinear elliptic equation describing the pressure of the saturating fluid (flowing through its pores). In this model, the permeability depends nonlinearly on the dilatation (divergence of the displacement) of the medium. We show that the solution has W 2 m regularity. We describe the numerical approximation of solutions using a hybrid finite element-least squares mixed finite element method. Error estimates are obtained through the introduction of an auxiliary linear elasticity equation. Numerical experiments verify the error estimates and validate the proposed poroelasticity model.

Research paper thumbnail of A First Order Scheme for Backward Doubly Stochastic Differential Equations

SIAM/ASA Journal on Uncertainty Quantification, 2016

In this paper, a fully discrete finite difference method for forward backward doubly stochastic d... more In this paper, a fully discrete finite difference method for forward backward doubly stochastic differential equation is studied. A first order numerical algorithm is obtained using the two sided Itô-Taylor expansion. Numerical experiments verify the accuracy and efficiency of the numerical algorithm. A nonlinear tracking problem is simulated using the first order algorithm.