arXiv subjects

Bhupen Deka

Publications and source records attributed to Bhupen Deka.

2 recordsLinked to original sources

arXiv2026

Numerical analysis of the second-order time-dependent saddle point Maxwell system via a parameter-free discontinuous Galerkin method: The first optimal ${\bf L}^{2}$-norm error estimates

We present a novel parameter-free discontinuous Galerkin (dG) finite element method (FEM) for the time-dependent Maxwell system formulated as a saddle point problem. We establish the stability of the proposed semi-discrete problem and derive optimal error estimates in energy and $ {\bf L}^{2} $ norms for the electric field variable, as well as in $ L^{2} $ norm for the potential function. To the best of our knowledge, this work provides the first optimal $ {\bf L}^{2} $-norm error analysis for the second-order time-dependent saddle point Maxwell equations using any variants of FEMs. Additionally, we propose several complete discrete time-integrators and verify the optimal convergence results through examples in both 2D and 3D setups.

math.NA↗

arXiv2021

A Systematic Study on Weak Galerkin Finite Element Method for Second Order Parabolic Problems

A systematic numerical study on weak Galerkin (WG) finite element method for second order linear parabolic problems is presented by allowing polynomial approximations with various degrees for each local element. Convergence of both semidiscrete and fully discrete WG solutions are established in $L^{\infty}(L^2)$ and $L^{\infty}(H^1)$ norms for a general WG element $({\cal P}_{k}(K),\;{\cal P}_{j}(\partial K),\;\big[{\cal P}_{l}(K)\big]^2)$, where $k\ge 1$, $j\ge 0$ and $l\ge 0$ are arbitrary integers. The fully discrete space-time discretization is based on a first order in time Euler scheme. Our results are intended to extend the numerical analysis of WG methods for elliptic problems [J. Sci. Comput., 74 (2018), 1369-1396] to parabolic problems. Numerical experiments are reported to justify the robustness, reliability and accuracy of the WG finite element method.

math.NA↗