arXiv subjects

Srinivasan Natesan

Publications and source records attributed to Srinivasan Natesan.

5 recordsLinked to original sources

arXiv2026

Error Analysis of a Fully-Discrete Implicit $\theta$-Scheme with WG-FEM for Parabolic Singularly Perturbed Boundary Turning Point Problems

In this article, we introduce a weak Galerkin finite element method (WG-FEM) for a class of parabolic singularly perturbed boundary turning point problems (SPBTPPs). The proposed numerical scheme employs an implicit $\theta$-scheme for temporal discretization over a uniform mesh and applies WG-FEM spatial discretization on a layer-adapted Shishkin mesh. Rigorous stability estimates are established for both the semi-discrete and fully-discrete formulations. Furthermore, we derive error estimates in the energy norm and prove that the convergence of the scheme is uniform with respect to the perturbation parameter. Numerical tests are conducted to verify the theoretical findings and illustrate the efficiency of the proposed method. In addition, the theoretical framework developed in this work lays the foundation for future extensions to higher-dimensional problems using ADI-type operator splitting WG-FEM schemes.

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arXiv2026

Physics-informed post-processing of stabilized finite element solutions for transient convection-dominated problems

The numerical simulation of convection-dominated transient transport phenomena poses significant computational challenges due to sharp gradients and propagating fronts across the spatiotemporal domain. Classical discretization methods often generate spurious oscillations, requiring advanced stabilization techniques. However, even stabilized finite element methods may require additional regularization to accurately resolve localized steep layers. On the other hand, standalone physics-informed neural networks (PINNs) struggle to capture sharp solution structures in convection-dominated regimes and typically require a large number of training epochs. This work presents a hybrid computational framework that extends the PINN-Augmented SUPG with Shock-Capturing (PASSC) methodology from steady to unsteady problems. The approach combines a semi-discrete stabilized finite element method with a PINN-based correction strategy for transient convection-diffusion-reaction equations. Stabilization is achieved using the Streamline-Upwind Petrov-Galerkin (SUPG) formulation augmented with a YZbeta shock-capturing operator. Rather than training over the entire space-time domain, the neural network is applied selectively near the terminal time, enhancing the finite element solution using the last K_s temporal snapshots while enforcing residual constraints from the governing equations and boundary conditions. The network incorporates residual blocks with random Fourier features and employs progressive training with adaptive loss weighting. Numerical experiments on five benchmark problems, including boundary and interior layers, traveling waves, and nonlinear Burgers dynamics, demonstrate significant accuracy improvements at the terminal time compared to standalone stabilized finite element solutions.

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arXiv2025

A Parameter-Driven Physics-Informed Neural Network Framework for Solving Two-Parameter Singular Perturbation Problems Involving Boundary Layers

In this article, our goal is to solve two-parameter singular perturbation problems (SPPs) in one- and two-dimensions using an adapted Physics-Informed Neural Networks (PINNs) approach. Such problems are of major importance in engineering and sciences as it appears in control theory, fluid and gas dynamics, financial modelling and so on. Solutions of such problems exhibit boundary and/or interior layers, which make them difficult to handle. It has been validated in the literature that standard PINNs have low accuracy and can't handle such problems efficiently. Recently Cao et. al \cite{cao2023physics} proposed a new parameter asymptotic PINNs (PA-PINNs) to solve one-parameter singularly perturbed convection-dominated problems. It was observed that PA-PINNs works better than standard PINNs and gPINNs in terms of accuracy, convergence and stability. In this article, for the first time robustness of PA-PINNs will be validated for solving two-parameter SPPs.

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arXiv2024

Anisotropic Error Analysis of Weak Galerkin finite element method for Singularly Perturbed Biharmonic Problems

We consider the Weak Galerkin finite element approximation of the Singularly Perturbed Biharmonic elliptic problem on a unit square domain with clamped boundary conditions. Shishkin mesh is used for domain discretization as the solution exhibits boundary layers near the domain boundary. Error estimates in the equivalent $H^{2}-$ norm have been established and the uniform convergence of the proposed method has been proved. Numerical examples are presented corroborating our theoretical findings.

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arXiv2023

Superconvergence analysis of interior penalty discontinuous Galerkin method for a class of time-fractional diffusion problems

In this study, we consider a class of non-autonomous time-fractional partial advection-diffusion-reaction (TF-ADR) equations with Caputo type fractional derivative. To obtain the numerical solution of the model problem, we apply the non-symmetric interior penalty Galerkin (NIPG) method in space on a uniform mesh and the L1-scheme in time on a graded mesh. It is demonstrated that the computed solution is discretely stable. Superconvergence of error estimates for the proposed method are obtained using the discrete energy-norm. Also, we have applied the proposed method to solve semilinear problems after linearizing by the Newton linearization process. The theoretical results are verified through numerical experiments.

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