Dibyendu Adak

Rujeko Chinomona, Dibyendu Adak, William J. Barham et al.

We present a tensor-train discontinuous Galerkin (TT-DG) formulation for the Vlasov--Maxwell system that combines a modal DG discretization with low-rank tensor representations of the phase-space solution and discrete operators. The formulation exploits the tensor-product structure of the DG discretization to perform quadrature, differentiation, nonlinear upwind flux evaluation, and time integration directly in compressed form. The method is evaluated on several standard 1D2V Vlasov--Maxwell benchmark problems, including the streaming Weibel instability, weak Landau damping, and two-stream instability problems. Across these problems, the TT formulation reproduces the accuracy and conservation behavior of the underlying full-grid DG discretization while substantially reducing memory usage and runtime. For weakly nonlinear problems, compression ratios exceeding $10^4$ are obtained together with significant speedups relative to the full-grid solver. For the strongly nonlinear two-stream instability problem, the TT formulation remains effective despite reduced compressibility caused by fine-scale phase-space filamentation. These results demonstrate that tensor-train representations provide an effective approach for reducing the computational cost of deterministic DG-based kinetic plasma simulations while retaining the favorable numerical properties of the underlying discretization.

Dibyendu Adak, E. Natarajan

We discuss nonconforming virtual element method for convection dominated (diffusive coefficient is very small compared to convective coefficient and reac- tion coefficient ) convection-diffusion-reaction equation using L^2 projection operator.In this paper we stabilize the stabilization terms using same technique which is used for stabilization of symmetric part in VEM, where T is an arbitrary element, and assume H^2(T) regularity on each element to prove polynomial consistency where v_h is approximate solution.We have shown that linear nonconforming VE is not convergent for convection dominated convection-diffusion reaction problem and higher regu larity of f ,source term is also needed for convergence analysis.The novelty of this paper is we introduce a new SDFEM type nonconforming virtual element method for convection-dominated convection diffusion reaction equation, and discuss the computability issue using degrees of freedom of element without explicit knowledge of basis functions of virtual element methods.The present framework is stable in the limit of vanishing diffusion.

Dibyendu Adak, E. Natarajan

In this paper we discuss the application of nonconforming virtual element methods(VEM) for the second order diffusion dominated convection diffusion reaction equation. Stability of the virtual element methods has been proved for the symmetric bilinear form. But the same analysis cannot be carried out for the non-symmetric case. In this work we present the external virtual element methods using $L^2$ projection operator and prove the well-posedness of VEM for non symmetric bilinear form. We also proved polynomial consistency of discrete bilinear form assuming $H^2$ regularity of approximate solution on each triangle. We have shown optimal convergence estimate in the broken sobolev norm.

Dibyendu Adak, E. Natarajan

It is shown in this paper that non-conforming finite elements on the triangle using $P^{1}$-nonconforming polynomials and $P^{2}$ -conforming polynomials can be easily built and used.They appear as an 'enriched' version of the standard piecewise quadratic six-node element.This work is divided into two parts.In the first we present the basic- property of the element,namely how it can be built and basic error estimates.We have observed that this new element behaves like $P^{1}$ non-conforming element.In the second part we have applied our element to the elliptic problem and the theoretical estimate has been guaranteed by numerical result.