Browsing by Author "Yadav, Sangita"

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Backward euler method for 2d sobolev equation with burgers’ type non-linearity
(AIP, 2023-06) Yadav, Sangita
Backward Euler for two dimensional Sobolev equation is discussed in this article. We begin by obtaining some basic a priori estimates for the semi-discrete scheme and for the backward Euler approximation. It is proven that these estimations for the discrete scheme are valid uniformly in time using the discrete Gronwall’s Lemma. In addition, the presence of a discrete global attractor is established. Furthermore, optimal a priori error bounds are determined, which are time dependent exponentially. Under the uniqueness condition, these error estimates are demonstrated to be uniform in time. Finally, we establish several numerical examples that validate our theoretical approach.
A conforming virtual element method for parabolic integro-differential equations
(De Gruyter, 2023-10) Yadav, Sangita
This article develops and analyses a conforming virtual element scheme for the spatial discretization of parabolic integro-differential equations combined with backward Euler’s scheme for temporal discretization. With the help of Ritz–Voltera and L2 projection operators, optimal a priori error estimates are established. Moreover, several numerical experiments are presented to confirm the computational efficiency of the proposed scheme and validate the theoretical findings.
Convergence analysis of virtual element methods for the Sobolev equation with convection
(Springer, 2025-07) Yadav, Sangita
We explore the potential applications of virtual elements for solving the Sobolev equation with a convective term. A conforming virtual element method is employed for spatial discretization, while an implicit Euler scheme is used to approximate the time derivative. To establish the optimal rate of convergence, a novel intermediate projection operator is introduced. We discuss and analyze both the semi-discrete and fully discrete schemes, deriving optimal error estimates for both the energy norm and -norm. Several numerical experiments are conducted to validate the theoretical findings and assess the computational efficiency of the proposed numerical methods.
Entropy generation and heat transfer in nonlinear Buoyancy–driven Darcy–Forchheimer hybrid nanofluids with activation energy
(De Gruyter, 2025-04) Sharma, Bhupendra Kumar; Yadav, Sangita
This study investigates the influence of a magnetic field, activation energy, and heat source on the heat and mass transfer within a cross fluid embedded with mono-, di-, and tri-nanoparticles, considering thermal radiation and Darcy–Forchheimer effects. Utilizing the Cattaneo–Christov theory, non-Fourier heat transfer is modeled for a vertical moving surface. A mathematical model is developed and subsequently converted into a dimensionless form through an appropriate similarity transformation, resulting in a system of first-order ordinary differential equations. The numerical approach to solve the system is BVP4C solver in MATLAB, a tool specifically designed for boundary value problems. Graphical representations have been analyzed for velocity profiles, temperature profiles, and concentration distributions for different values of physical parameters. It is observed that the velocity profiles exhibit an upward trend with an increase in the parameters associated with nonlinear thermal convection and nonlinear concentration convection. Additionally, the analysis of surface shear stress, heat transfer coefficients, and mass transfer coefficients revealed that an increase in the porosity parameter and Forchheimer number results in decreased shear stress. Entropy generation is also investigated to quantify irreversibilities in the system. The analysis showed that increasing the Brinkman number, diffusion parameter, and temperature and concentration difference parameters leads to higher entropy generation, indicating greater irreversibility in the system. A comparative analysis demonstrates that tri-nanoparticles substantially improve flow velocity, thermal conductivity, and solute diffusion compared to di- and mono-nanoparticles, with tri-nanofluids exhibiting the most optimal overall performance.
Hdg method for linear parabolic integro-differential equations
(Elsevier, 2023-08) Yadav, Sangita
This paper develops the hybridizable discontinuous Galerkin (HDG) method for a linear parabolic integro-differential equation and analyzes uniform in time error bounds. To handle the integral term, an extended Ritz-Volterra projection is introduced, which helps in achieving optimal order convergence of for the semi-discrete problem when polynomials of degree are used to approximate both the solution and the flux variables. Further, element-by-element post-processing is proposed, and it is established that it achieves convergence of the order for . Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.
Hdg method for nonlinear parabolic integro-differential equations
(De Gruyter, 2024-04) Yadav, Sangita
The hybridizable discontinuous Galerkin (HDG) method has been applied to a nonlinear parabolic integro-differential equation. The nonlinear functions are considered to be Lipschitz continuous to analyze uniform in time a priori bounds. An extended type Ritz–Volterra projection is introduced and used along with the HDG projection as an intermediate projection to achieve optimal order convergence of O(hk+1) when polynomials of degree k≥0 are used to approximate both the solution and the flux variables. By relaxing the assumptions in the nonlinear variable, super-convergence is achieved by element-by-element post-processing. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.
An hp-local Discontinuous Galerkin Method for Parabolic Integro-Differential Equations
(Springer, 2020-06) Yadav, Sangita
In this article, a priori error bounds are derived for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that error estimates in L 2-norm of the gradient as well as of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps us to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains.
Hybridizable discontinuous Galerkin method for nonlinear hyperbolic integro-differential equations
(Elsevier, 2025-08) Yadav, Sangita
In this paper, we present the hybridizable discontinuous Galerkin (HDG) method for a nonlinear hyperbolic integro-differential equation. We discuss the semi-discrete and fully-discrete error analysis of the method. For the semi-discrete error analysis, an extended type mixed Ritz-Volterra projection is introduced for the model problem. It helps to achieve the optimal order of convergence for the unknown scalar variable and its gradient. Further, a local post-processing is performed, which helps to achieve super-convergence. Subsequently, by employing the central difference scheme in the temporal direction and applying the mid-point rule for discretizing the integral term, a fully discrete scheme is formulated, accompanied by its corresponding error estimates. Ultimately, through the examination of numerical examples within two-dimensional domains, computational findings are acquired, thus validating the results of our study.
Hybridizable discontinuous galerkin method for strongly damped wave problem
(Springer, 2025-01) Yadav, Sangita
We introduce and analyze a hybridizable discontinuous Galerkin (HDG) approach for the strongly damped linear wave equation. In our investigation, we derive a priori error estimates to demonstrate the optimal convergence of the approximations for both the solution and its gradient. Further, with the help of the dual problem, we present a post-processed solution and analyze its convergence rate, which is of order for , where k is the degree of the polynomial. We also propose a fully discrete scheme, which is of . To validate our theoretical findings, we perform numerical experiments.
Legendre wavelet modified petrov–galerkin method in two-dimensional moving boundary problem
(De Gruyter, 2017-12) Yadav, Sangita
In this study, we developed the two-dimensional Legendre wavelet modified Petrov–Galerkin method for solving the two-dimensional moving boundary problem arising during melting of solid whose one surface is kept under most generalised boundary condition, and other two surfaces are insulated. The particular cases when surface subjected to the boundary condition of first, second and third kinds are discussed in detail. For validity of the present method, we have plotted graphs between residual (obtained from the original differential equation and its associated boundary conditions) and x-axis and found the effect of an error on moving layer thickness and y coordinate, respectively. Furthermore, we proved the convergence analysis of present method. The effect of parameters (Predvoditelev number, Kirpichev number, Biot number) on the moving layer thickness is discussed in detail. The whole analysis is presented in a dimensionless form.
Mixed virtual element method for integro-differential equations of parabolic type
(Springer, 2024-04) Yadav, Sangita
This article presents and analyzes a mixed virtual element approach for discretizing parabolic integro-differential equations in a bounded subset of , in addition to the backward Euler approach for temporal discretization. With the help of the intermediate projection along with Fortin and projections, we effectively tackle the treatment of integral terms in both the fully discrete and semi-discrete analysis. This inclusion leads to the derivation of optimal a priori error estimates with an order of for the two unknowns. Furthermore, we present a systematic analysis that outlines the step-by-step process for achieving super convergence of the discrete solution, with an order of . Several computational experiments are discussed to validate the proposed scheme’s computational efficiency and support the theoretical conclusions.
Mixed virtual element method for linear parabolic integro-differential equations
(Global Science Press, 2024) Yadav, Sangita
This article develops and analyses a mixed virtual element scheme for the spatial discretization of linear parabolic integro-differential equations (PIDEs) combined with backward Euler’s temporal discretization approach. The introduction of mixed Ritz-Volterra projection significantly helps in managing the integral terms, yielding optimal convergence of order O(hk+1) for the two unknowns p(x,t) and σ(x,t). In addition, a step-by-step analysis is proposed for the super convergence of the discrete solution of order O(hk+2). The fully discrete case has also been analyzed and discussed to achieve O(τ) in time. Several computational experiments are discussed to validate the proposed schemes computational efficiency and support the theoretical conclusions
Modified Method of Characteristics Combined with Finite Volume Element Methods for Incompressible Miscible Displacement Problems in Porous Media
(Hindawi Publishing Corporation, 2014) Yadav, Sangita
The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure-velocity equation and the concentration equation. In this paper, we present a mixed finite volume element method (FVEM) for the approximation of the pressure-velocity equation. Since modified method of characteristics (MMOC) minimizes the grid orientation effect, for the approximation of the concentration equation, we apply a standard FVEM combined with MMOC. A priori error estimates in norm are derived for velocity, pressure and concentration. Numerical results are presented to substantiate the validity of the theoretical results.
On two conservative hdg schemes for nonlinear klein-gordon equation
(2024-11) Yadav, Sangita
In this article, a hybridizable discontinuous Galerkin (HDG) method is proposed and analyzed for the Klein-Gordon equation with local Lipschitz-type non-linearity. {\it A priori} error estimates are derived, and it is proved that approximations of the flux and the displacement converge with order O(hk+1), where h is the discretizing parameter and k is the degree of the piecewise polynomials to approximate both flux and displacement variables. After post-processing of the semi-discrete solution, it is shown that the post-processed solution converges with order O(hk+2) for k≥1. Moreover, a second-order conservative finite difference scheme is applied to discretize in time %second-order convergence in time. and it is proved that the discrete energy is conserved with optimal error estimates for the completely discrete method. %Since at each time step, one has to solve a nonlinear system of algebraic equations, To avoid solving a nonlinear system of algebraic equations at each time step, a non-conservative scheme is proposed, and its error analysis is also briefly established. Moreover, another variant of the HDG scheme is analyzed, and error estimates are established. Finally, some numerical experiments are conducted to confirm our theoretical findings.
Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data
(Springer, 2013-05) Yadav, Sangita
In the first part of this article, a new mixed method is proposed and analyzed for parabolic integro-differential equations (PIDE) with nonsmooth initial data. Compared to the standard mixed method for PIDE, the present method does not bank on a reformulation using a resolvent operator. Based on energy arguments combined with a repeated use of an integral operator and without using parabolic type duality technique, optimal L2-error estimates are derived for semidiscrete approximations, when the initial condition is in L2. Due to the presence of the integral term, it is, further, observed that a negative norm estimate plays a crucial role in our error analysis. Moreover, the proposed analysis follows the spirit of the proof techniques used in deriving optimal error estimates for finite element approximations to PIDE with smooth data and therefore, it unifies both the theories, i.e., one for smooth data and other for nonsmooth data. Finally, we extend the proposed analysis to the standard mixed method for PIDE with rough initial data and provide an optimal error estimate in L2, which improves upon the results available in the literature.
Optimal L2 estimates for semidiscrete Galerkin methods for parabolic integro-differential equations with nonsmooth data
(OUP, 2014) Yadav, Sangita
In this article, we discuss an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth initial data. It is based on energy arguments and on a repeated use of time integration, but without using parabolic type duality technique. Optimal L2- error estimate is derived for the semidiscrete approximation, when the initial data is in L2.
Optimal L2 estimates for the semidiscrete galerkin method applied to parabolic integro-differential equations with nonsmooth data
(CUP, 2024-06) Yadav, Sangita
We propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal L2-error estimate is derived for the semidiscrete approximation when the initial data is in L2. A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain
A priori error estimates for sobolev equation using HDG method
(Springer, 2025-08) Yadav, Sangita
A hybridizable discontinuous Galerkin (HDG) method is introduced and analyzed to solve the Sobolev equation. The analysis includes the derivation of a priori error estimates, demonstrating that the approximations for both the flux and displacement exhibit convergence at a rate of order where h represents the mesh size and k is the polynomial degree. Additionally, the solution is further improved by applying a post-processing technique, and it has been demonstrated that, for , the post-processed solution converges at an enhanced rate of order . A fully discrete scheme is also proposed, achieving second-order accuracy in time; numerical results are needed to validate the theoretical results.
A priori hp-estimates for discontinuous Galerkin approximations to linear hyperbolic integro-differential equations
(Elsevier, 2015-10) Yadav, Sangita
An hp-discontinuous Galerkin (DG) method is applied to a class of second order linear hyperbolic integro-differential equations. Based on the analysis of an expanded mixed type Ritz–Volterra projection, a priori hp-error estimates in -norm of the velocity as well as of the displacement, which are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p are derived. For optimal estimates of the displacement in -norm with reduced regularity on the exact solution, a variant of Baker's nonstandard energy formulation is developed and analyzed. Results on order of convergence which are similar in spirit to linear elliptic and parabolic problems are established for the semidiscrete case after suitably modifying the numerical fluxes. For the completely discrete scheme, an implicit-in-time procedure is formulated, stability results are derived and a priori error estimates are discussed. Finally, numerical experiments on two dimensional domains are conducted which confirm the theoretical results.
Superconvergent discontinuous galerkin methods for nonlinear elliptic equations
(American Mathematical Society, 2013-07) Yadav, Sangita
Based on the analysis of Cockburn et al. [Math. Comp. 78 (2009), pp. 1-24] for a selfadjoint linear elliptic equation, we first discuss superconvergence results for nonselfadjoint linear elliptic problems using discontinuous Galerkin methods. Further, we have extended our analysis to derive superconvergence results for quasilinear elliptic problems. When piecewise polynomials of degree k ≥ 1 are used to approximate both the potential as well as the flux, it is shown, in this article, that the error estimate for the discrete flux in L2-norm is of order k + 1. Further, based on solving a discrete linear elliptic problem at each element, a suitable postprocessing of the discrete potential is developed and then, it is proved that the resulting post-processed potential converges with order of convergence k + 2 in L2-norm. These results confirm superconvergent results for linear elliptic problems.