arXiv subjects

Arbaz Khan

Publications and source records attributed to Arbaz Khan.

At least 19 recordsLinked to original sources

arXiv2026

Mixed Hybrid High-Order Methods for Sixth-Order Problems: Fully Non-conforming and $C^0$-Conforming Discretizations

We consider a class of sixth-order elliptic partial differential equations in two and three dimensions subject to simply supported and Cahn--Hilliard-type boundary conditions. Based on the Ciarlet--Raviart reformulation, we recast the sixth-order problem as an equivalent mixed system involving second- and fourth-order equations and establish its well-posedness under suitable assumptions. For the resulting mixed formulation, we propose and analyse two discretization frameworks: a fully non-conforming hybrid high-order (HHO) method on general polytopal meshes and a $C^0$-conforming HHO--finite element method on simplicial meshes. We prove stability and derive optimal-order a priori error estimates for the primary variables under appropriate regularity assumptions. For the fully non-conforming HHO method, we further derive a reliable residual-based a posteriori error estimator. Numerical experiments confirm the theoretical convergence rates and illustrate the robustness of the proposed methods for a range of mobility parameters.

math.NA↗

arXiv2026

A locking free mixed FEM based on a pure pseudostress based formulation for the elasticity eigenproblem

We analyze a novel locking-free mixed formulation for the elasticity eigenvalue problem in both two and three dimensions, expressed exclusively in terms of the pseudostress tensor. An important feature of this formulation is that it does not require the enforcement of symmetry, either in a weak or strong sense. The displacement of the structure is recovered via a postprocess of the computed pseudostress. We introduce a mixed finite element method based in the tensorial version of the standard families of finite elements to discretize the space $\boldsymbol{\mathcal{H}}(\bdiv)$. We prove convergence and a priori error estimates under the theory of non-compact operators. Additionally, we perform an a posteriori error analysis for the problem, proving reliability and efficiency of the proposed indicator. We validate our theoretical results with numerical tests on different geometrical and physical configurations.

math.NA↗

arXiv2026

Nitsche-based FEM for the Laplace eigenvalue problem: spectral approximation and a posteriori error analysis

In this paper, we present the numerical analysis of an elliptic eigenvalue problem in which the essential boundary condition is imposed weakly by means of the Nitsche method. The resulting discrete eigenvalue problem is studied within the framework of compact operator theory. We prove norm convergence of the discrete solution operator and derive error estimates for the eigenvalues and eigenfunctions, with rates depending on the chosen Nitsche variant. In addition, we develop an a posteriori error analysis and propose a residual-based estimator suitable for adaptive refinement. Several numerical experiments are presented to assess the convergence, stability and robustness of the method, including the influence of the Nitsche stabilization parameter and the performance of the adaptive strategy.

math.NA↗

arXiv2026

Distributed optimal control problems governed by poroelasticity equations

In this paper, we propose and analyze a novel two-field symmetric formulation with solid displacement and fluid pressure as main unknowns for the Biot's consolidation model in poroelasticity. Firstly, we prove the well-posedness of the new formulation and then show the existence and uniqueness of optimal control where the fluid sources in the model act as a control variable. We prove a priori error estimates for the fully discrete scheme with backward Euler time discretization and a variational approximation of the control variable. A numerical example is presented to validate the performance of the proposed novel scheme.

math.OC↗

arXiv2026

Structure-Preserving and Pressure-Robust PINNs for Incompressible Oseen Problems

We develop a new class of physics-informed neural network approximations for the stationary Oseen equations based on stability-consistent loss constructions. In contrast to standard PINN formulations, which are typically heuristic, the proposed consistent PINN (CPINN) framework is systematically derived from the stability structure of the continuous problem. Within this setting, we introduce two fundamentally new approaches. First, we design standard CPINN formulations that exhibit clear improvements over conventional PINNs. Second, we propose pressure-robust CPINN formulations that provably eliminate the influence of gradient forces on the velocity approximation, yielding velocity errors that depend solely on the divergence-free component of the forcing and are independent of the pressure. The framework accommodates both exactly divergence-free architectures and unconstrained velocity approximations, providing a unified treatment of these two paradigms. Using techniques from optimal recovery theory, we establish, for the first time in the PINN setting for Oseen-type problems, quantitative recovery estimates and optimal error bounds for both velocity and pressure under suitable Besov regularity assumptions. In particular, we obtain optimal rates for the velocity in $\boldsymbol{H}^1(\Omega)$ and for the pressure in $L^2(\Omega)$. The proposed methodology introduces a pressure-robust CPINN paradigm for incompressible flows, combining structural consistency, robustness with respect to irrotational forces, and rigorous accuracy guarantees. Numerical experiments corroborate the theoretical findings and demonstrate the effectiveness of the approach.

math.NA↗

arXiv2026

Mixed Consistent PINNs for Elliptic Obstacle Problems with Stability Analysis

We propose a consistent physics-informed neural networks (CPINNs) framework for elliptic obstacle problems formulated as variational inequalities. The method is based on a mixed loss functional that is rigorously aligned with the stability structure of the underlying problem and incorporates obstacle constraints through a consistent treatment of the associated Lagrange multiplier. Relying on optimal recovery theory under Besov regularity assumptions, we establish near-optimal convergence rates for the simultaneous reconstruction of the solution and the multiplier from pointwise interior and boundary data. To enable practical implementation, we construct discrete counterparts of the continuous stability norms and duality pairings, leading to fully computable and theoretically justified training losses. Numerical experiments on benchmark obstacle problems demonstrate the accuracy, stability, and robustness of the proposed approach, and highlight its clear advantages over standard PINNs.

math.NA↗

arXiv2025

Finite element analysis for a Herrmann pressure formulation of the elastoacoustic problem with variable coefficients

In two and three dimensions, this study is focused on the numerical analysis of an eigenproblem associated with a fluid-structure model for sloshing and elasto-acoustic vibration. We use a displacement-Herrmann pressure formulation for the solid, while for the fluid, a pure displacement formulation is considered. Under this approach we propose a non conforming locking-free method based on classic finite elements to approximate the natural frequencies (of the eigenmodes) of the coupled system. Employing the theory for non-compact operators we prove convergence and error estimates. Also we propose an a posteriori error estimator for this coupled problem which is shown to be efficient and reliable. All the presented theory is contrasted with a set of numerical tests in 2D and 3D.

math.NA↗

arXiv2025

A priori error analysis of consistent PINNs for parabolic PDEs

We present a new a priori analysis of a class of collocation methods for parabolic PDEs that rely only on pointwise data of force term, boundary data, and initial data. Under Besov regularity assumptions, we characterize the optimal recovery rate of the solution u based on sample complexity. We establish error bounds by constructing a new consistent loss function that effectively controls the approximation error. This loss incorporates contributions from the interior, boundary, and initial data in a discretized form and is designed to reflect the true PDE structure. Our theoretical results demonstrate that minimizing this loss function yields near optimal recovery under suitable conditions of regularity and sampling. Novel practical variants of the loss function are discussed, and numerical experiments confirm the effectiveness.

math.NA↗

arXiv2025

Dynamic output-based feedback stabilizability for linear parabolic equations with memory

The stabilizability of a general class of linear parabolic equations with a memory term, is achieve by explicit output feedback. The control input is given as a function of a state-estimate provided by an exponential dynamic Luenberger observer based on the output of sensor measurements. The numbers of actuators and sensors are finite. The feedback input and output injection operators are given explicitly involving appropriate orthogonal projections. For exponential kernels, exponential stabilizability can be achieved with the rate of the exponential kernel. The discretization and simulation of the controlled systems are addressed as well and results of simulations are reported showing the performance of the proposed dynamic output-based control feedback input. We include simulations for both exponential and weakly singular Riesz kernels, showing the success of the strategy in obtaining a stabilizing input.

math.OC↗

arXiv2024

2022 Flood Impact in Pakistan: Remote Sensing Assessment of Agricultural and Urban Damage

Pakistan was hit by the world's deadliest flood in June 2022, causing agriculture and infrastructure damage across the country. Remote sensing technology offers a cost-effective and efficient method for flood impact assessment. This study is aimed to assess the impact of flooding on crops and built-up areas. Landsat 9 imagery, European Space Agency-Land Use/Land Cover (ESA-LULC) and Soil Moisture Active Passive (SMAP) data are used to identify and quantify the extent of flood-affected areas, crop damage, and built-up area destruction. The findings indicate that Sindh, a province in Pakistan, suffered the most. This impact destroyed most Kharif season crops, typically cultivated from March to November. Using the SMAP satellite data, it is assessed that the high amount of soil moisture after flood also caused a significant delay in the cultivation of Rabi crops. The findings of this study provide valuable information for decision-makers and stakeholders involved in flood risk management and disaster response.

cs.CY↗

arXiv2024

A priori and a posteriori error bounds for the fully mixed FEM formulation of poroelasticity with stress-dependent permeability

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. The problem is adequately structured into a coupled system consisting of one saddle-point formulation, one linearised perturbed saddle-point formulation, and two off-diagonal perturbations. This system's unique solvability requires assumptions on regularity and Lipschitz continuity of the inverse permeability, and the analysis follows fixed-point arguments and the Babu\v{s}ka--Brezzi theory. The discrete problem is shown uniquely solvable by applying similar fixed-point and saddle-point techniques as for the continuous case. The method is based on the classical PEERS$_k$ elements, it is exactly momentum and mass conservative, and it is robust with respect to the nearly incompressible as well as vanishing storativity limits. We derive a priori error estimates, we also propose fully computable residual-based a posteriori error indicators, and show that they are reliable and efficient with respect to the natural norms, and robust in the limit of near incompressibility. These a posteriori error estimates are used to drive adaptive mesh refinement. The theoretical analysis is supported and illustrated by several numerical examples in 2D and 3D.

math.NA↗

arXiv2024

hp-discontinuous Galerkin method for the generalized Burgers-Huxley equation with weakly singular kernels

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 the local mesh size, polynomial degree, and solution regularity, achieving optimal convergence in the energy norm. For the fully-discrete scheme, we initially implement the $hp$-finite element method (conforming), followed by the $hp$-discontinuous Galerkin method. We establish the well-posedness and stability of the fully-discrete scheme and provide corresponding a priori estimates. The effectiveness of the proposed method is demonstrated through numerical validation on a series of test problems.

math.NA↗

arXiv2024

Conforming/Non-Conforming Mixed Finite Element Methods for Optimal Control of Velocity-Vorticity-Pressure Formulation for the Oseen Problem with Variable Viscosity

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 both the constitutive equation and the incompressibility condition, is well-posed under certain assumptions on the viscosity parameter. The CG method is divergence-conforming and suits any Stokes inf-sup stable velocity-pressure finite element pair, while a generic discrete space approximates vorticity. The DG scheme employs a stabilization technique, and a piecewise constant discretization estimates the control variable. We establish optimal a priori and residual-based a posteriori error estimates for the proposed schemes. Finally, we provide numerical experiments to showcase the method's performance and effectiveness.

math.NA↗

arXiv2024

Divergence conforming DG method for the optimal control of the Oseen equation with variable viscosity

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 divergence-conforming DGFEM approach. Moreover, we thoroughly analyze $L^2$ error estimates for scenarios dominated by diffusion and convection. Additionally, we establish the new reliable and efficient a posteriori error estimators for the optimal control of the Oseen equation with variable viscosity. Theoretical findings are validated through numerical experiments conducted in both two and three dimensions.

math.NA↗

arXiv2024

Non-Conforming Structure Preserving Finite Element Method for Doubly Diffusive Flows on Bounded Lipschitz Domains

We study a stationary model of doubly diffusive flows with temperature-dependent viscosity on bounded Lipschitz domains in two and three dimensions. A new well-posedness and regularity analysis of weak solutions under minimal assumptions on domain geometry and data regularity are established. A fully non-conforming finite element method based on Crouzeix-Raviart elements, which ensures locally exactly divergence-free velocity fields is explored. Unlike previously proposed schemes, this discretization enables to establish uniqueness of the discrete solutions. We prove the well-posedness of the discrete problem and derive a priori error estimates. An accuracy test is conducted to verify the theoretical error decay rates in flow, Stokes and Darcy regimes on convex and non-convex domains, and a benchmark test of flow in a porous cavity is conducted, comparing the proposed method with existing literature.

math.NA↗

arXiv2024

A posteriori error estimates for the Generalized Burgers-Huxley equation with weakly singular kernels

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 discontinuous Galerkin finite element method (DGFEM) in spatial dimensions. Additionally, employing backward Euler and Crank Nicolson discretization in the temporal domain and DGFEM in spatial dimensions, we introduce an estimator for the fully discrete GBHE, taking into account the influence of past history. The paper also establishes optimal $L^2$ error estimates for both the stationary GBHE and GBHE. Ultimately, we validate the effectiveness of the proposed error estimator through numerical results, demonstrating its efficacy in an adaptive refinement strategy.

math.NA↗

arXiv2024

Interior penalty discontinuous Galerkin methods for the nearly incompressible elasticity eigenvalue problem with heterogeneous media

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 problems as the mesh size approaches zero, independently of the Lamé constants. Additionally, we conduct an a posteriori analysis and propose a reliable and efficient estimator. The theoretical findings are supported by numerical experiments.

math.NA↗

arXiv2024

Divergence conforming finite element methods for flow-transport coupling with osmotic effects

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 transport on the membrane, while mixed boundary conditions are considered in the remainder of the boundary. We show existence and uniqueness of the continuous problem using a fixed-point argument. Next, an H(div)-conforming finite element formulation is proposed, and we address its a priori error analysis. The method uses an upwind approach that provides stability in the convection-dominated regime. We showcase a set of numerical examples validating the theory and illustrating the use of the new methods in the simulation of reverse osmosis processes.

math.NA↗