Devika Shylaja
@iist.ac.in
Postdoctoral Scholar
Indian Institute of Space Science and Technology Trivandum
RESEARCH INTERESTS
Numerical Analysis, Finite Element Methods
Scopus Publications
Scopus Publications
Carsten Carstensen, Neela Nataraj, Gopikrishnan C. Remesan, and Devika Shylaja
Springer Science and Business Media LLC
AbstractA unified framework for fourth-order semilinear problems with trilinear nonlinearity and general sources allows for quasi-best approximation with lowest-order finite element methods. This paper establishes the stability and a priori error control in the piecewise energy and weaker Sobolev norms under minimal hypotheses. Applications include the stream function vorticity formulation of the incompressible 2D Navier-Stokes equations and the von Kármán equations with Morley, discontinuous Galerkin, $$C^{0}$$ C 0 interior penalty, and weakly over-penalized symmetric interior penalty schemes. The proposed new discretizations consider quasi-optimal smoothers for the source term and smoother-type modifications inside the nonlinear terms.
Sudipto Chowdhury, Asha K. Dond, Neela Nataraj, and Devika Shylaja
EDP Sciences
This article discusses the numerical analysis of the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain in ℝ2. The state and adjoint variables are discretised using the nonconforming Morley finite element method and the control is discretized using piecewise constant functions. A priori and a posteriori error estimates are derived for the state, adjoint and control variables. The a posteriori error estimates are shown to be efficient. Numerical results that confirm the theoretical estimates are presented.
M Thamban Nair and Devika Shylaja
IOP Publishing
Abstract This paper deals with the numerical approximation of the biharmonic inverse source problem in an abstract setting in which the measurement data is finite-dimensional. This unified framework in particular covers the conforming and nonconforming finite element methods (FEMs). The inverse problem is analysed through the forward problem. Error estimate for the forward solution is derived in an abstract set-up that applies to conforming and Morley nonconforming FEMs. Since the inverse problem is ill-posed, Tikhonov regularization is considered to obtain a stable approximate solution. Error estimate is established for the regularized solution for different regularization schemes. Numerical results that confirm the theoretical results are also presented.
Carsten Carstensen, Sharat Gaddam, Neela Nataraj, Amiya K. Pani, and Devika Shylaja
EDP Sciences
This paper focusses on the von Kármán equations for the moderately large deformation of a very thin plate with the convex obstacle constraint leading to a coupled system of semilinear fourth-order obstacle problem and motivates its nonconforming Morley finite element approximation. The first part establishes the well-posedness of the von Kármán obstacle problem and also discusses the uniqueness of the solution under an a priori and an a posteriori smallness condition on the data. The second part of the article discusses the regularity result of Frehse from 1971 and combines it with the regularity of the solution on a polygonal domain. The third part of the article shows an a priori error estimate for optimal convergence rates for the Morley finite element approximation to the von Kármán obstacle problem for small data. The article concludes with numerical results that illustrates the requirement of smallness assumption on the data for optimal convergence rate.
Jérome Droniou, Neela Nataraj, and Devika Shylaja
Springer Science and Business Media LLC
This paper deals with the Hessian discretisation method (HDM) for fourth order semi-linear elliptic equations with a trilinear nonlinearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as, the conforming and non-conforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth order semi-linear elliptic equations with trilinear nonlinearity. Four properties namely, the coercivity, consistency, limit-conformity and compactness enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications namely, the Navier--Stokes equations in stream function vorticity formulation and the von Karman equations of plate bending are discussed. Results of numerical experiments are presented for the Morley ncFEM and GR method.
Sudipto Chowdhury, Neela Nataraj, and Devika Shylaja
Walter de Gruyter GmbH
Abstract Consider the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain of ℝ 2 {\\mathbb{R}^{2}} that describe the deflection of very thin plates with box constraints on the control variable. This article discusses a numerical approximation of the problem that employs the Morley nonconforming finite element method (FEM) to discretize the state and adjoint variables. The control is discretized using piecewise constants. A priori error estimates are derived for the state, adjoint and control variables under minimal regularity assumptions on the exact solution. Error estimates in lower-order norms for the state and adjoint variables are derived. The lower-order estimates for the adjoint variable and a post-processing of control leads to an improved error estimate for the control variable. Numerical results confirm the theoretical results obtained.
Devika Shylaja
Wiley
The Hessian discretisation method (HDM) for fourth order linear elliptic equations provides a unified convergence analysis framework based on three properties namely coercivity, consistency, and limit-conformity. Some examples that fit in this approach include conforming and nonconforming finite element methods, finite volume methods and methods based on gradient recovery operators. A generic error estimate has been established in $L^2$, $H^1$ and $H^2$-like norms in literature. In this paper, we establish improved $L^2$ and $H^1$ error estimates in the framework of HDM and illustrate it on various schemes. Since an improved $L^2$ estimate is not expected in general for finite volume method (FVM), a modified FVM is designed by changing the quadrature of the source term and a superconvergence result is proved for this modified FVM. In addition to the Adini nonconforming finite element method (ncFEM), in this paper, we show that the Morley ncFEM is an example of HDM. Numerical results that justify the theoretical results are also presented.
Jérôme Droniou, Bishnu P. Lamichhane, and Devika Shylaja
Springer Science and Business Media LLC
In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is applied to linear fourth order problems. It is shown that HDM encompasses a large number of numerical methods for fourth order elliptic problems: finite element methods (conforming and non-conforming) as well as finite volume methods. We also use the HDM to design a novel method, based on conforming $$\\mathbb {P}_1$$P1 finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.
Jérome Droniou, Neela Nataraj, and Devika Shylaja
Walter de Gruyter GmbH
Abstract The article discusses the gradient discretisation method (GDM) for distributed optimal control problems governed by diffusion equation with pure Neumann boundary condition. Using the GDM framework enables to develop an analysis that directly applies to a wide range of numerical schemes, from conforming and non-conforming finite elements, to mixed finite elements, to finite volumes and mimetic finite differences methods. Optimal order error estimates for state, adjoint and control variables for low-order schemes are derived under standard regularity assumptions. A novel projection relation between the optimal control and the adjoint variable allows the proof of a super-convergence result for post-processed control. Numerical experiments performed using a modified active set strategy algorithm for conforming, non-conforming and mimetic finite difference methods confirm the theoretical rates of convergence.
Jérome Droniou, Neela Nataraj, and Devika Shylaja
Society for Industrial & Applied Mathematics (SIAM)
In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretization method. Gradient schemes are defined for the optimality system of the control problem. Error estimates for state, adjoint, and control variables are derived. Superconvergence results for gradient schemes under realistic regularity assumptions on the exact solution are discussed. These superconvergence results are shown to apply to nonconforming $\\mathbb{P}_1$ finite elements and to the mixed-hybrid mimetic finite differences. Results of numerical experiments are demonstrated for the conforming, nonconforming and mixed-hybrid mimetic finite difference schemes.