arXiv subjects

Dibyendu Adak

Publications and source records attributed to Dibyendu Adak.

15 recordsLinked to original sources

arXiv2026

A Tensor-Train Discontinuous Galerkin Method for the Vlasov-Maxwell System

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.

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arXiv2026

Conforming and non-conforming virtual element methods for the biharmonic Steklov eigenvalue problem with minimum regularity

In this work, we analyze the conforming and $C^0$-non-conforming Virtual Element Method for a fourth-order Steklov eigenvalue problem on a generally shaped, possibly nonconvex, polygonal domain. By employing an {\it enriching } operator, we derive the convergence analysis using the discrete $H^2$ seminorm, and the $H^1$ and $L^2$ norms. We use the Babuška--Osborn spectral theory \cite{BO} to prove that the numerical scheme approximates the spectrum without introducing any spurious eigenvalue. Moreover, we derive the optimal order of convergence for eigenfunctions and double order for eigenvalues. We assess the performance of the method on several numerical tests using different families of polygonal meshes.

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arXiv2026

Nonconforming virtual element methods for fourth-order nonlinear reaction-diffusion systems: a unified framework and analysis

We develop a unified framework for the design and analysis of high-order nonconforming virtual element methods for nonlinear fourth-order reaction--diffusion problems in two dimensions, with emphasis on clamped, Navier, and Cahn--Hilliard-type boundary conditions. Time discretization is performed using the backward Euler scheme, while the spatial approximation relies on nonconforming virtual element spaces of arbitrary order $k \ge 2$, encompassing both $C^0$-nonconforming and Morley-type methods. A key contribution of this work is the development of a novel and rigorous unified error analysis for these numerical schemes, applicable to domains that are not necessarily convex, differing from the existing literature. By introducing a class of Companion operators, we construct novel Ritz-type projections and derive a new error equation that enables us to obtain optimal error estimates for the scheme under a minimal spatial regularity assumption on the weak solution. Finally, we present numerical experiments on polygonal meshes as applications of the proposed framework, including the extended Fisher--Kolmogorov equation, and a fourth-order model with Cahn--Hilliard-type boundary conditions, which validate the theoretical results and illustrate the performance of the method for the three classes of boundary conditions.

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arXiv2025

Space-Time Spectral Collocation Tensor-Network Approach for Maxwell's Equations

In this work, we develop a space--time Chebyshev spectral collocation method for three-dimensional Maxwell's equations and combine it with tensor-network techniques in Tensor-Train (TT) format. Under constant material parameters, the Maxwell system is reduced to a vector wave equation for the electric field, which we discretize globally in space and time using a staggered spectral collocation scheme. The staggered polynomial spaces are designed so that the discrete curl and divergence operators preserve the divergence-free constraint on the magnetic field. The magnetic field is then recovered in a space--time post-processing step via a discrete version of Faraday's law. The global space--time formulation yields a large but highly structured linear system, which we approximate in low-rank TT-format directly from the operator and data, without assuming that the forcing is separable in space and time. We derive condition-number bounds for the resulting operator and prove spectral convergence estimates for both the electric and magnetic fields. Numerical experiments for three-dimensional electromagnetic test problems confirm the theoretical convergence rates and show that the TT-based solver maintains accuracy with approximately linear complexity in the number of grid points in space and time.

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arXiv2024

A noncoforming virtual element approximation for the Oseen eigenvalue problem

In this paper we analyze a nonconforming virtual element method to approximate the eigenfunctions and eigenvalues of the two dimensional Oseen eigenvalue problem. The spaces under consideration lead to a divergence-free method which is capable to capture properly the divergence at discrete level and the eigenvalues and eigenfunctions. Under the compact theory for operators we prove convergence and error estimates for the method. By employing the theory of compact operators we recover the double order of convergence of the spectrum. Finally, we present numerical tests to assess the performance of the proposed numerical scheme.

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arXiv2024

Space-Time Spectral Element Tensor Network Approach for Time Dependent Convection Diffusion Reaction Equation with Variable Coefficients

In this paper, we present a new space-time Petrov-Galerkin-like method. This method utilizes a mixed formulation of Tensor Train (TT) and Quantized Tensor Train (QTT), designed for the spectral element discretization (Q1-SEM) of the time-dependent convection-diffusion-reaction (CDR) equation. We reformulate the assembly process of the spectral element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the spectral element operators. Recognizing the banded structure inherent in the spectral element framework's discrete operators, we further exploit the QTT format of the CDR to achieve greater speed and compression. Additionally, we present a comprehensive approach for integrating variable coefficients of CDR into the global discrete operators within the TT/QTT framework. The effectiveness of the proposed method, in terms of memory efficiency and computational complexity, is demonstrated through a series of numerical experiments, including a semi-linear example.

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arXiv2024

Tensor Network Space-Time Spectral Collocation Method for Solving the Nonlinear Convection Diffusion Equation

Spectral methods provide highly accurate numerical solutions for partial differential equations, exhibiting exponential convergence with the number of spectral nodes. Traditionally, in addressing time-dependent nonlinear problems, attention has been on low-order finite difference schemes for time discretization and spectral element schemes for spatial variables. However, our recent developments have resulted in the application of spectral methods to both space and time variables, preserving spectral convergence in both domains. Leveraging Tensor Train techniques, our approach tackles the curse of dimensionality inherent in space-time methods. Here, we extend this methodology to the nonlinear time-dependent convection-diffusion equation. Our discretization scheme exhibits a low-rank structure, facilitating translation to tensor-train (TT) format. Nevertheless, controlling the TT-rank across Newton's iterations, needed to deal with the nonlinearity, poses a challenge, leading us to devise the "Step Truncation TT-Newton" method. We demonstrate the exponential convergence of our methods through various benchmark examples. Importantly, our scheme offers significantly reduced memory requirement compared to the full-grid scheme.

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arXiv2024

Tensor Network Space-Time Spectral Collocation Method for Time Dependent Convection-Diffusion-Reaction Equations

Emerging tensor network techniques for solutions of Partial Differential Equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultrafast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for solution of the time dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time dependent PDEs often use finite difference for time, and a spectral scheme for the spatial dimensions, which leads to slow linear convergence. Spectral collocation space-time methods show exponential convergence, however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to very high compression of linear operators from terabytes to kilobytes in TT-format, and tens of thousands times speedup when compared to full grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

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arXiv2023

A $C^1$-conforming arbitrary-order two-dimensional virtual element method for the fourth-order phase-field equation

We present a two-dimensional conforming virtual element method for the fourth-order phase-field equation. Our proposed numerical approach to the solution of this high-order phase-field (HOPF) equation relies on the design of an arbitrary-order accurate, virtual element space with $C^1$ global regularity. Such regularity is guaranteed by taking the values of the virtual element functions and their full gradient at the mesh vertices as degrees of freedom. Attaining high-order accuracy requires also edge polynomial moments of the trace of the virtual element functions and their normal derivatives. In this work, we detail the scheme construction, and prove its convergence by deriving error estimates in different norms. A set of representative test cases allows us to assess the behavior of the method.

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arXiv2022

VEM approximation for the Stokes eigenvalue problem: a priori and a posteriori error analysis

The present paper proposes an inf-sup stable divergence free virtual element method and associated a priori, and a posteriori error analysis to approximate the eigenvalues and eigenfunctions of the Stokes spectral problem in one shot. For the a priori analysis, we take advantage of the compactness of the solution operator to prove convergence of the eigenfunctions and double order convergence of eigenvalues. Additionally we also propose an a posteriori estimator of residual type, which we prove is reliable and efficient, in order to perform adaptive refinements that allow to recover the optimal order of convergence for non smooth eigenfunctions. A set of representative numerical examples investigates such theoretical results.

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arXiv2021

Virtual element approximation of two-dimensional parabolic variational inequalities

We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Theorem. The mass matrix is computed through an approximate L 2 polynomial projection, whose properties are carefully investigated in the paper. We prove the well-posedness of the resulting scheme in two different ways that reveal the contractive nature of the VEM and its connection with the minimization of quadratic functionals. The convergence analysis requires the existence of a nonnegative quasi-interpolation operator, whose construction is also discussed in the paper. The variational crime introduced by the virtual element setting produces five error terms that we control by estimating a suitable upper bound. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time on three different mesh families including distorted squares, nonconvex elements, and Voronoi tesselations.

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arXiv2019

Virtual element method for semilinear sine-Gordon equation over polygonal mesh using product approximation technique

In this paper, we employ the linear virtual element spaces to discretize the semilinear sine-Gordon equation in two dimensions. The salient features of the virtual element method (VEM) are: (a) it does not require explicit form of the shape functions to construct the nonlinear and the bilinear terms, and (b) relaxes the constraint on the mesh topology by allowing the domain to be discretized with general polygons consisting of both convex and concave elements, and (c) easy mesh refinements (hanging nodes and interfaces are allowed). The nonlinear source term is discretized by employing the product approximation technique and for temporal discretization, the Crank-Nicolson scheme is used. The resulting nonlinear equations are solved using the Newton's method. We derive a priori error estimations in $L^2$ and $H^1$ norms. The convergence properties and the accuracy of the virtual element method for the solution of the sine-Gordon equation are demonstrated with academic numerical experiments.

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arXiv2016

A Unified Analysis of Nonconforming Virtual Element Methods for Convection Diffusion Reaction Problem

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.

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arXiv2015

Analysis of nonconforming virtual element method for the convection diffusion reaction equation with polynomial coefficients

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.

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arXiv2015

Analysis of a combined NC1-C2 method for elliptic problem

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.

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