Achyuta Ranjan Dutta Mohapatra

Achyuta Ranjan Dutta Mohapatra, Bhupen Deka

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.

Achyuta Ranjan Dutta Mohapatra

We are concerned in designing a suitable numerical scheme based on the equal-order hybrid high-order (HHO) method for the linear parabolic integro-differential equations. The spatial discretization is made using the equal-order HHO method and subsequently we perform the stability analysis of the corresponding semi-discrete scheme. The convergence results are presented in suitably defined Bochner norms for the semi-discrete problem. Then a second-order temporal discretization is implemented on the time domain using a Crank-Nicolson scheme where the memory term is approximated using a composite trapezoidal quadrature rule. The stability of the resultant complete discrete schemes are analyzed followed by derivation of the error estimates of order $\mathcal{O}(τ^{2}+h^{k+1})$, $k\ge 0$ is the degree of local polynomial approximation, in discrete $l^{2}(0,T;H^{1}(Ω))$ and $l^{\infty}(0,T;H^{1}(Ω))$ like norms. Numerical illustrations are performed on some polygonal meshes validating the theoretical estimates.