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Introduction
Arbaz Khan currently works at the Department of Mathematics, Indian Institute of Technology Roorkee (IITR). Arbaz does research in Applied Mathematics.
Additional affiliations
January 2020 - present
June 2017 - January 2020
January 2015 - August 2015
Education
July 2008 - January 2015
Publications
Publications (53)
The results of a recent extension of the analysis of an ‐conforming method for a model of double‐diffusive flow in porous media introduced in [Bürger, Méndez, Ruiz‐Baier, SINUM (2019), 57:1318–1343] to the time‐dependent case are summarized. These include the efficiency and reliability of residual‐based a posteriori error estimators for the steady,...
In this study, we explore the theoretical and numerical aspects of the generalized Burgers-Huxley equation (a non-linear advection-diffusion-reaction problem) incorporating weakly singular kernels in a d-dimensional domain, where . For the continuous problem, we provide an in-depth discussion on the existence and the uniqueness of weak solution usi...
In this paper, the numerical approximation of the generalized Burgers'-Huxley equation (GBHE) with weakly singular kernels using non-conforming methods will be presented. Specifically, we discuss two new formulations. The first formulation is based on the non-conforming finite element method (NCFEM). The other formulation is based on discontinuous...
We develop a family of mixed finite element methods for a model of nonlinear poroelasticity where, thanks to a rewriting of the constitutive equations, the permeability depends on the total poroelastic stress and on the fluid pressure and therefore we can use the Hellinger-Reissner principle with weakly imposed stress symmetry for Biot's equations....
In this work, we investigate the $hp$-discontinuous Galerkin (DG) time-stepping method for the generalized Burgers-Huxley equation with memory, a non-linear advection-diffusion-reaction problem featuring weakly singular kernels. We derive a priori error estimates for the semi-discrete scheme using $hp$-DG time-stepping, with explicit dependence on...
This work examines the distributed optimal control of generalized Oseen equations with non-constant viscosity. We propose and analyze a new conforming augmented mixed finite element method and a Discontinuous Galerkin (DG) method for the velocity-vorticity-pressure formulation. The continuous formulation, which incorporates least-squares terms from...
This study introduces the divergence-conforming discontinuous Galerkin finite element method (DGFEM) for numerically approximating optimal control problems with distributed constraints, specifically those governed by stationary generalized Oseen equations. We provide optimal a priori error estimates in energy norms for such problems using the diver...
In this work, we propose fully nonconforming, locally exactly divergence-free discretizations based on lowest order Crouziex-Raviart finite element and piecewise constant spaces to study the optimal control of stationary double diffusion model presented in [Bu ̈rger, M ́endez, Ruiz-Baier, SINUM (2019), 57:1318-1343]. The well-posedness of the discr...
This paper explores the residual based a posteriori error estimations for the generalized Burgers-Huxley equation (GBHE) featuring weakly singular kernels. Initially, we present a reliable and efficient error estimator for both the stationary GBHE and the semi-discrete GBHE with memory, utilizing the discontin-uous Galerkin finite element method (D...
Mixed finite element methods Divergence-conforming schemes A priori error analysis A posteriori error analysis Operator preconditioning We present a finite element discretization to model the interaction between a poroelastic structure and an elastic medium. The consolidation problem considers fully coupled deformations across an interface, ensurin...
This paper studies the family of interior penalty discontinuous Galerkin methods for solving the Herrmann formulation of the linear elasticity eigenvalue problem in heterogeneous media. By employing a weighted Lamé coefficient norm within the framework of non-compact operators theory, we prove convergence of both continuous and discrete eigenvalue...
We propose a model for the coupling of flow and transport equations with porous membrane-type conditions on part of the boundary. The governing equations consist of the incompressible Navier--Stokes equations coupled with an advection-diffusion equation, and we employ a Lagrange multiplier to enforce the coupling between penetration velocity and tr...
In this paper we discuss the optimal convergence of a standard adaptive
scheme based on mixed finite element approximation to the solution of the
eigenvalue problem associated with the Stokes equations. The proofs of the
quasi-orthogonality and the discrete reliability are presented. Our numerical
experiments confirm the efficacy of the proposed ad...
In this paper we present a mathematical and numerical analysis of an eigenvalue problem associated to the elasticity-Stokes equations stated in two and three dimensions. Both problems are related through the Herrmann pressure. Employing the Babuška-Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are w...
The analysis of an $\textbf {H}(\textrm {div})$-conforming method for a model of double-diffusive flow in porous media introduced in Bürger, Méndez & Ruiz-Baier (2019, On H(div)-conforming methods for double-diffusion equations in porous media. SIAM J. Numer. Anal., 57,1318–1343) is extended to the time-dependent case. In addition, the efficiency a...
The analysis of a delayed generalized Burgers-Huxley equation (a non-linear advection-diffusion-reaction problem) with weakly singular kernels is carried out in this work. Moreover, numerical approximations are performed using the conforming finite element method (CFEM). The existence, uniqueness and regularity results for the continuous problem ha...
In this work, a theoretical framework is developed to study the control constrained distributed optimal control of a stationary double diffusion model presented in [Burger, Mendez, Ruiz-Baier, SINUM (2019), 57:1318-1343]. For the control problem, as the source term belongs to a weaker space, a new solvability analysis of the governing equation is p...
We present a finite element discretisation to model the interaction between a poroelastic structure and an elastic medium. The consolidation problem considers fully coupled deformations across an interface, ensuring continuity of displacement and total traction, as well as no-flux for the fluid phase. Our formulation of the poroelasticity equations...
The objective of the conference is to bring forth the proactive researchers working in the discipline of Dynamical Systems, Mathematical Control Theory, Stochastic Differential Equations, Inverse Problems and their applications in Neural Network, Robotics, etc. in science and engineering. In this conference, the renowned Indian Mathematician Profes...
In this work we address the analysis of the stationary generalized Burgers-Huxley equation (a nonlinear elliptic problem with anomalous advection) and propose conforming, nonconforming and discontinuous Galerkin finite element methods for its numerical approximation. The existence, uniqueness and regularity of weak solutions are discussed in detail...
We develop the \textit{a posteriori} error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. The discretisations use $H^1$-conforming finite elements of degree $k+1$ for displacement and fluid pressure, and discontinuous piecewise polynomials of...
The analysis of the double-diffusion model and $\mathbf{H}(\mathrm{div})$-conforming method introduced in [B\"urger, M\'endez, Ruiz-Baier, SINUM (2019), 57:1318--1343] is extended to the time-dependent case. In addition, the efficiency and reliability analysis of residual-based {\it a posteriori} error estimators for the steady, semi-discrete, and...
In this work we address the analysis of the stationary generalized Burgers-Huxley equation (a nonlinear elliptic problem with anomalous advection) and propose conforming, nonconforming and discontinuous Galerkin finite element methods for its numerical approximation. The existence, uniqueness and regularity of weak solutions is discussed in detail...
This work is dedicated to the memory of John W. Barrett, who introduced the concept of inf–sup stability to the corresponding author in the bar at the MAFELAP conference in 1981.
We analyze a posteriori error estimators for locking-free mixed finite element approximation of Biot’s consolidation model. Three estimators are described. The simplest of...
We consider the system of partial differential equations stemming from the time discretization of the two-field formulation of the Biot's model with the backward Euler scheme. A typical difficulty encountered in the space discretization of this problem is the robustness with respect to various material parameters. We deal with this issue by observi...
Linear poroelasticity models have a number of important applications in biology and geophysics. In particular, Biot's consolidation model is a well-known model that describes the coupled interaction between the linear response of a porous elastic medium and a diffusive fluid flow within it, assuming small deformations. Although deterministic linear...
In this paper, we present a divergence-conforming discontinuous Galerkin finite element method for Stokes eigenvalue problems. We prove a priori error estimates for the eigenvalue and eigenfunction errors and present a residual based a posteriori error estimator. The a posteriori error estimator is proven to be reliable and (locally) efficient. We...
In this paper, a least-squares spectral method and a non-conforming least-squares spectral element method for three dimensional linear elliptic system will be presented. Differentiability estimates and the main stability theorem for the proposed method are proven. Using the regularity estimate and the proposed stability estimates, we introduce a su...
The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field PDE model in which the Young's modulus is an affine function of a countable set of parameters. We analyse the weak formulation, its stability with respect to a weighte...
This paper is devoted to studying the Arnold-Winther mixed finite element method for two-dimensional Stokes eigenvalue problems using the stress-velocity formulation. A priori error estimates for the eigenvalue and eigenfunction errors are presented. To improve the approximation for both eigenvalues and eigenfunctions, we propose a local postproces...
In this paper, we present a divergence-conforming discontinuous Galerkin finite element method for Stokes eigenvalue problems. We prove a priori error estimates for the eigenvalue and eigenfunction errors and present a robust residual based a posteriori error estimator. The a posteriori error estimator is proven to be reliable and (locally) efficie...
We consider the nearly incompressible linear elasticity problem with an uncertain spatially varying Young's modulus. The uncertainty is modelled with a finite set of parameters with prescribed probability distribution. We introduce a novel three-field mixed variational formulation of the PDE model and discuss its approximation by stochastic Galerki...
We consider the nearly incompressible linear elasticity problem with an uncertain spatially varying Young's modulus. The uncertainty is modelled with a finite set of parameters with prescribed probability distribution. We introduce a novel three-field mixed variational formulation of the PDE model and discuss its approximation by stochastic Galerki...
In this paper, an h∕p spectral element method with least-square formulation for parabolic interface problem will be presented. The regularity result of the parabolic interface problem is proven for non-homogeneous interface data. The differentiability estimates and the main stability estimate theorem, using non-conforming spectral element functions...
We consider so-called Herrmann and Hydrostatic mixed formulations of classical linear elasticity and analyse the error associated with locally stabilised $P_1-P_0$ finite element approximation. First, we prove a stability estimate for the discrete problem and establish an a priori estimate for the associated energy error. Second, we consider a resi...
This paper is devoted to study the Arnold-Winther mixed finite element method for two dimensional Stokes eigenvalue problems using the stress-velocity formulation. A priori error estimates for the eigenvalue and eigenfunction errors are presented. To improve the approximation for both eigenvalues and eigenfunctions, we propose a local post-processi...
In this paper, a least-squares spectral element method for parabolic initial value problem for two space dimension on parallel computers is presented. The theory is also valid for three dimension. This method gives exponential accuracy in both space and time. The method is based on minimization of residuals in terms of the partial differential equa...
This paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic solid is almost incompressible. Several novel a posteriori error estimators for the energy norm of the finite element error are proposed and analysed. We establish upper a...
This paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic solid is almost incompressible. Several novel a posteriori error estimators for the energy norm of the finite element error are proposed and analysed. We establish upper a...
In this paper, we propose preconditioners for the system of linear equations that arise from a discretization of fourth order elliptic problems in two and three dimensions (d=2,3) using spectral element methods. These preconditioners are constructed using separation of variables and can be diagonalized and hence easy to invert. For second order ell...
In this paper, a fully non-conforming least-squares spectral element method for fourth order elliptic problems on smooth domains is presented. The proposed method works for a general fourth order elliptic operator with non-homogeneous data in two dimensions and gives exponentially accurate solutions. We derive differentiability estimates and prove...
In this paper we propose a least-squares spectral element method for three dimensional elliptic interface problems. The differentiability estimates and the main stability theorem, using non-conforming spectral element functions, are proven. The proposed method is free from any kind of first order reformulation. A suitable preconditioner is construc...
Non-conforming approximation methods are becoming increasingly popular because of the potential to apply to multi-material and multi-model analysis for both bounded and unbounded domains. In this paper, we present a least-square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain. The method gives...
Several methods have been proposed in the literature for solving the Black–Scholes equation for European Options. The method proposed in the current study achieves spectral accuracy in both space and time. The method is based on minimization of a functional given in terms of the sum of squares of the residuals in the partial differential equation a...
Our experimental study shows that in the case of strongly absorbing samples, the time resolved thermal lens (TL) signal has a distinctive new feature. Existing TL models that are only based on heat conduction cannot explain this behavior. We have therefore developed a more comprehensive model that can also explain this additional feature in the TL...