arXiv subjects

Gouranga Mallik

Publications and source records attributed to Gouranga Mallik.

12 recordsLinked to original sources

arXiv2026

A Hybrid High-Order Finite Element Method for a Nonlocal Nonlinear Problem of Kirchhoff Type

In this article, we design and analyze a hybrid high-order (HHO) finite element approximation for the solution of a nonlocal nonlinear problem of Kirchhoff type. The HHO method involves arbitrary-order polynomial approximations on structured and unstructured polytopal meshes. We establish the existence of a unique discrete solution to the nonlocal nonlinear discrete problem. We derive an optimal-order error estimate in the discrete energy norm. The discrete system is solved using Newton's iterations on the sparse matrix system. We perform numerical tests to substantiate the theoretical results.

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arXiv2025

An Error Analysis of Second Order Elliptic Optimal Control Problem via Hybrid Higher Order Methods

This paper presents the design and analysis of a Hybrid High-Order (HHO) approximation for a distributed optimal control problem governed by the Poisson equation. We propose three distinct schemes to address unconstrained control problems and two schemes for constrained control problems. For the unconstrained control problem, while standard finite elements achieve a convergence rate of $ k+1 $ (with $ k $ representing the polynomial degree), our approach enhances this rate to $ k+2 $ by selecting the control from a carefully constructed reconstruction space. For the box-constrained problem, we demonstrate that using lowest-order elements ($ \mathbb{P}_0 $) yields linear convergence, in contrast to finite element methods (FEM) that require linear elements to achieve comparable results. Furthermore, we derive a cubic convergence rate for control in the variational discretization scheme. Numerical experiments are provided to validate the theoretical findings.

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arXiv2024

A Local Projection Stabilised HHO Method for the Oseen Problem

In this article, we consider a local projection stabilisation for a Hybrid High-Order (HHO) approximation of the Oseen problem. We prove an existence-uniqueness result under a stronger SUPG-like norm. We improve the stability and provide error estimation in stronger norm for convection dominated Oseen problem. We also derive an optimal order error estimate under the SUPG-like norm for equal-order polynomial discretisation of velocity and pressure spaces. Numerical experiments are performed to validate the theoretical results.

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arXiv2023

A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems

In this article, we design and analyze a Hybrid High-Order (HHO) finite element approximation for a class of strongly nonlinear boundary value problems. We consider an HHO discretization for a suitable linearized problem and show its well-posedness using the Gardings type inequality. The essential ingredients for the HHO approximation involve local reconstruction and high-order stabilization. We establish the existence of a unique solution for the HHO approximation using the Brouwer fixed point theorem and contraction principle. We derive an optimal order a priori error estimate in the discrete energy norm. Numerical experiments are performed to illustrate the convergence histories.

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arXiv2021

Finite element analysis of the Dirichlet boundary control problem governed by linear parabolic equation

A finite element analysis of a Dirichlet boundary control problem governed by the linear parabolic equation is presented in this article. The Dirichlet control is considered in a closed and convex subset of the energy space $H^1(Ω\times(0,T)).$ We prove well-posedness and discuss some regularity results for the control problem. We derive the optimality system for the optimal control problem. The first order necessary optimality condition results in a simplified Signorini type problem for control variable. The space discretization of the state variable is done using conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. To discretize the control we use the conforming prismatic Lagrange finite elements. We derive an optimal order of convergence of error in control, state, and adjoint state. The theoretical results are corroborated by some numerical tests.

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arXiv2021

A Hybrid-High Order Method for Quasilinear Elliptic Problems of Nonmonotone Type

In this paper, we design and analyze a Hybrid-High Order (HHO) approximation for a class of quasilinear elliptic problems of nonmonotone type. The proposed method has several advantages, for instance, it supports arbitrary order of approximation and general polytopal meshes. The key ingredients involve local reconstruction and high-order stabilization terms. Existence and uniqueness of the discrete solution are shown by Brouwer's fixed point theorem and contraction result. A priori error estimate is shown in discrete energy norm that shows optimal order convergence rate. Numerical experiments are performed to substantiate the theoretical results.

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arXiv2021

Goal-Oriented A Posteriori Error Estimation for the Biharmonic Problem Based on Equilibrated Moment Tensor

In this article, goal-oriented a posteriori error estimation for the biharmonic plate bending problem is considered. The error for approximation of goal functional is represented by an estimator which combines dual-weighted residual method and equilibrated moment tensor. An abstract unified framework for the goal-oriented a posteriori error estimation is derived. In particular, $C^0$ interior penalty and discontinuous Galerkin finite element methods are employed for practical realization. The abstract estimation is based on equilibrated moment tensor and potential reconstruction that provides a guaranteed upper bound for the goal error. Numerical experiments are performed to illustrate the effectivity of the estimators.

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arXiv2020

Comparison results of $P_2$-finite elements for fourth-order semilinear von Karman equations

Lower-order $P_2$ finite elements are popular for solving fourth-order elliptic PDEs when the solution has limited regularity. A priori and a posteriori error estimates for von Karman equations are considered in Carstensen et al. (2019, 2020) with respect to different mesh dependent norms which involve different jump and penalization terms. This paper addresses the question, whether they are comparable with respect to a common norm. This article establishes that the errors for the quadratic symmetric interior discontinuous Galerkin, $C^0$ interior penalty and nonconforming Morley finite element methods are equivalent upto some higher-order oscillation term with respect to a unified norm. Numerical experiments are performed to substantiate the comparison results.

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arXiv2019

Nonconforming Finite Element Discretisation for Semilinear Problems with Trilinear Nonlinearity

The Morley finite element method (FEM) is attractive for semilinear problems with the biharmonic operator as a leading term in the stream function vorticity formulation of 2D Navier-Stokes problem and in the von Kármán equations. This paper establishes a best-approximation a~priori error analysis and an a~posteriori error analysis of discrete solutions close to an arbitrary regular solution on the continuous level to semilinear problems with a trilinear nonlinearity. The analysis avoids any smallness assumptions on the data and so has to provide discrete stability by a perturbation analysis before the Newton-Kantorovic theorem can provide the existence of discrete solutions. An abstract framework for the stability analysis in terms of discrete operators from the medius analysis leads to new results on the nonconforming Crouzeix-Raviart FEM for second-order linear non-selfadjoint and indefinite elliptic problems with $L^\infty$ coefficients. The paper identifies six parameters and sufficient conditions for the local a~priori and a~posteriori error control of conforming and nonconforming discretisations of a class of semilinear elliptic problems first in an abstract framework and then in the two semilinear applications. This leads to new best-approximation error estimates and to a~posteriori error estimates in terms of explicit residual-based error control for the conforming and Morley FEM.

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arXiv2017

Error estimates for the numerical approximation of a distributed optimal control problem governed by the von Kármán equations

In this paper, we discuss the numerical approximation of a distributed optimal control problem governed by the von Karman equations, defined in polygonal domains with point-wise control constraints. Conforming finite elements are employed to discretize the state and adjoint variables. The control is discretized using piece-wise constant approximations. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Numerical results that justify the theoretical results are presented.

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arXiv2017

A Priori and A Posteriori Error Control of Discontinuous Galerkin Finite Element Methods for the Von Kármán Equations

This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von Kármán equations defined on a polygonal domain. A discrete inf-sup condition sufficient for the stability of the discontinuous Galerkin discretization of a well-posed linear problem is established and this allows the proof of local existence and uniqueness of a discrete solution to the non-linear problem with a Banach fixed point theorem. The Newton scheme is locally second-order convergent and appears to be a robust solution strategy up to machine precision. A comprehensive a priori and a posteriori energy-norm error analysis relies on one sufficiently large stabilization parameter and sufficiently fine triangulations. In case the other stabilization parameter degenerates towards infinity, the DGFEM reduces to a novel $C^0$ interior penalty method (IPDG). Moreover, a reliable and efficient a posteriori error analysis immediately follows for the DGFEM of this paper, while the different norms in the known $C^0$-IPDG lead to complications with some non-residual type remaining terms. Numerical experiments confirm the best-approximation results and the equivalence of the error and the error estimators. A related adaptive mesh-refining algorithm leads to optimal empirical convergence rates for a non convex domain.

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arXiv2015

A Nonconforming Finite Element Approximation for the von Karman Equations

In this paper, a nonconforming finite element method has been proposed and analyzed for the von Karman equations that describe bending of thin elastic plates. Optimal order error estimates in broken energy and $H^1$ norms are derived under minimal regularity assumptions. Numerical results that justify the theoretical results are presented.

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