Department of Mathematics

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Subject: search.filters.subject.Parabolic integro-differential equations

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    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
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    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.
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    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.
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    Two mixed virtual element formulations for parabolic integro-differential equations with nonsmooth initial data
    (Elsevier, 2025-03) Yadav, Sangita
    This article presents and examines two distinctive approaches to the mixed virtual element method (VEM) applied to parabolic integro-differential equations (PIDEs) with non-smooth initial data. In the first part of the paper, we introduce and analyze a mixed virtual element scheme for PIDE that eliminates the need for the resolvent operator. Through the introduction of a novel projection involving a memory term, coupled with the application of energy arguments and the repeated use of an integral operator, this study establishes optimal -error estimates for the two unknowns p and σ. Furthermore, optimal error estimates are derived for the standard mixed formulation with a resolvent kernel. The paper offers a comprehensive analysis of the VEM, encompassing both formulations.