s. gowrisankar

In this article, we explore the discontinuous Galerkin finite element method for two-parametric singularly perturbed convection-diffusion problems with a discontinuous source term. Due to the discontinuity in the source term, the problem typically shows a weak interior layer. Also, the presence of multiple perturbation parameters in the problem causes boundary layers on both sides of the boundary. In this work, we develop the nonsymmetric discontinuous Galerkin finite element method with interior penalties to handle the layer phenomenon. With the use of a typical Shishkin mesh, the domain is discretized, and a uniform error estimate is obtained. Numerical experiments are conducted to validate the theoretical conclusions.

In this paper, we focus on investigating a post-processing technique de-signed for one-dimensional singularly perturbed parabolic convection-diffusion problems that demonstrate a regular boundary layer. We use a back-ward Euler numerical approach for time derivatives with uniform mesh in the temporal direction, and a simple upwind scheme is used for spa-tial derivatives with modified graded mesh in the spatial direction. In this study, we demonstrate the effectiveness of the Richardson extrapola-tion technique in enhancing the ε-uniform accuracy of simple upwinding within the discrete supremum norm, as evidenced by an improvement from O(N −1 ln(1/ε) + △θ) to O(N −2 ln2(1/ε) + △θ2). Furthermore, to validate the theoretical findings, computational experiments are conducted for two test examples by applying the proposed technique.

In this article, we develop an efficient numerical method for one-dimensional time-delayed singularly perturbed parabolic problems. The proposed nu-merical approach comprises an upwind difference scheme with modified graded mesh in the spatial direction and a backward Euler scheme on uni-form mesh in the temporal direction. In order to capture the local behavior of the solutions, stability and error estimations are obtained with respect to the maximum norm. The proposed numerical method converges uniformly with first-order up to logarithm in the spatial variable and also first-order in the temporal variable. Finally, the outcomes of the numerical experiments are included for two test problems to validate the theoretical findings.