Gouranga Mallik

Gouranga Mallik
Indian Institute of Science | IISC · Department of Mathematics

Ph.D., IIT Bombay

About

13
Publications
2,369
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105
Citations
Introduction
Gouranga Mallik currently works at IISc Bangalore as a Raman postdoctoral researcher. Currently, he is working on Hybrid High-Order (HHO) approximation for PDEs. He worked on goal-oriented a posteriori error estimator for Darcy flow problem. During PhD, he worked on finite element approximation for the solution of semilinear fourth-order problem namely von Karman equations.
Additional affiliations
June 2017 - May 2018
IFP Energies nouvelles
Position
  • PostDoc Position
June 2011 - January 2017
Indian Institute of Technology Bombay
Position
  • PhD
Education
June 2011 - January 2017
Indian Institute of Technology Bombay
Field of study
  • Numerical Analysis
July 2009 - May 2011
July 2006 - June 2009
Ramakrishna Mission Vidyamandira
Field of study
  • Mathematics

Publications

Publications (13)
Preprint
Full-text available
In this article, we discuss goal-oriented a posteriori error estimation for the biharmonic plate bending problem. The error for a numerical approximation of a goal functional is represented by several computable estimators. One of these estimators is obtained using the dual-weighted residual method, which takes advantage of an equilibrated moment t...
Article
In this article, we discuss goal-oriented a posteriori error estimation for the biharmonic plate bending problem. The error for a numerical approximation of a goal functional is represented by several computable estimators. One of these estimators is obtained using the dual-weighted residual method, which takes advantage of an equilibrated moment t...
Preprint
Full-text available
A finite element analysis of a Dirichlet boundary control problem governed by the linear para-bolic equation is presented in this article. The Dirichlet control is considered in a closed and convex subset of the energy space H 1 (Ω × (0, T)). We prove well-posedness and discuss some regularity results for the control problem. We derive the optimali...
Preprint
Full-text available
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 stabilizat...
Preprint
Full-text available
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 pa...
Article
Full-text available
We derive a unified framework for goal-oriented a posteriori estimation covering in particular higher-order conforming, nonconforming, and discontinuous Galerkin finite element methods, as well as the finite volume method. The considered problem is a model linear second-order elliptic equation with inhomogeneous Dirichlet and Neumann boundary condi...
Article
Full-text available
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 approximati...
Article
Full-text available
This paper analyses discontinuous Galerkin finite element methods (DGFEM) to approximate a regular solution to the von K\'arm\'an 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 l...
Article
Full-text available
framework involves six parameters in (H1)-(H6) for the error control of conforming and nonconforming discretizations of a class of semilinear elliptic problems. The leading linear and elliptic differential operator, e.g., the Laplace or bi-Laplace, allows a low-order quadratic perturbation in the nonlinear problem. Provided the parameters in (H1)-...
Article
Full-text available
In this paper, we consider canonical von Kármán equations that describe the bending of thin elastic plates defined on polygonal domains. A conforming finite element method is employed to approximate the displacement and Airy stress functions. Optimal order error estimates in energy, $H^1$ and $L^2$ norms are deduced. The results of numerical experi...
Article
The canonical von Kármán equations which describe bending of thin elastic plates defined on polygonal domains are considered. Conforming and nonconforming finite element methods are employed to approximate the displacement and Airy stress functions. The results of the numerical experiments justify the theoretical results.
Article
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.

Questions

Questions (4)
Question
Does anyone have explicit C++ code to solve biharmonic problem ($\Delta^2 u=f$, subject to boundary conditions ) using any finite element methods?
Question
It seems that 'P3dc' element in FreeFem++ does not support third order derivatives in the code. The third order derivative is very necessary for the weak formulation.
Can someone please help me how to overcome these shortcomings?
Question
It seems that for a nonlinear map $f: \mathbb{R}^n \mapsto \mathbb{R}^n$ Newton's method solves (suppose f is sufficiantly smooth and solution exists) the nonlinear problem: for given $b$, find $a$ such that
$f(a)=b$.
with a initial guess $a_0$ sufficiently close to $a$.
Theoretically, the method converges quadratically to the exact solution $a$. But in actual numerical computation (Matlab etc.) if the number of unknowns $n$ is very large the method seems to be very slow and sometimes even do not converge quadratically. 
Is there any recent development to overcome this shortcomings?

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Projects

Projects (5)
Project
The aim of this project is to design and analyze Hybrid High-Order methods for PDEs and to obtain a robust numerical approximation. This includes polygonal/polytopal mesh and arbitrary order polynomial approximations.
Project
Numerical approximation and error estimation for optimal control problem governed by PDEs.