arXiv subjects

Devika Shylaja

Publications and source records attributed to Devika Shylaja.

14 recordsLinked to original sources

arXiv2026

Convergence Analysis of the Hessian Discretisation Method for Fourth Order Semi-linear Elliptic Equations with General Source

This paper presents a convergence analysis for the Hessian Discretisation Method (HDM) applied to fourth-order semilinear elliptic equations involving a trilinear nonlinearity and general source, based on two complementary approaches. The HDM serves as a unified framework for the convergence analysis of various numerical schemes, including conforming and nonconforming finite element methods (ncFEMs) and gradient recovery (GR) based methods. Error estimates for the Adini ncFEM and GR methods are derived for the first time, which provide an explicit order of convergence. The analysis relies on four key HDM properties along with a suitable companion operator to establish convergence results. Moreover, a convergence analysis is developed within the HDM framework, which does not require additional regularity assumptions on the exact solution or the assumption that the exact solution is regular. The paper further discusses two significant applications: the Navier--Stokes equations in stream function--vorticity formulation and the von K\'{a}rm\'{a}n equations for plate bending. Numerical experiments are provided to demonstrate the performance of the GR method, Morley, and Adini ncFEMs.

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arXiv2024

Convergence analysis of a nonconforming virtual element method for compressible miscible displacement problems in porous media

This article presents a priori error estimates of the miscible displacement of one compressible fluid by another in a porous medium. The study utilizes the $H(\rm div)$ conforming virtual element method (VEM) for the approximation of the velocity, while a non-conforming virtual element approach is employed for the concentration. The pressure is discretised using the standard piecewise discontinuous polynomial functions. These spatial discretization techniques are combined with a backward Euler difference scheme for time discretization. Error estimates are established for velocity, pressure and concentration. The article also includes numerical results that validate the theoretical estimates.

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arXiv2024

Nonconforming virtual element method for an incompressible miscible displacement problem in porous media

This article presents a priori error estimates of the miscible displacement of one incompressible fluid by another through a porous medium characterized by a coupled system of nonlinear elliptic and parabolic equations. The study utilizes the $H(\rm{div})$ conforming virtual element method (VEM) for the approximation of the velocity, while a non-conforming virtual element approach is employed for the concentration. The pressure is discretised using the standard piecewise discontinuous polynomial functions. These spatial discretization techniques are combined with a backward Euler difference scheme for time discretization. The article also includes numerical results that validate the theoretical estimates presented.

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arXiv2023

Morley Type Virtual Element Method for Von K\'{a}rm\'{a}n Equations

This paper analyses the nonconforming Morley type virtual element method to approximate a regular solution to the von K\'{a}rm\'{a}n equations that describes bending of very thin elastic plates. Local existence and uniqueness of a discrete solution to the non-linear problem is discussed. A priori error estimate in the energy norm is established under minimal regularity assumptions on the exact solution. Error estimates in piecewise $H^1$ and $L^2$ norm are also derived. A working procedure to find an approximation for the discrete solution using Newtons method is discussed. Numerical results that justify theoretical estimates are presented.

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arXiv2023

Unified a priori analysis of four second-order FEM for fourth-order quadratic semilinear problems

A unified framework for fourth-order semilinear problems with trilinear nonlinearity and general source 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 K\'{a}rm\'{a}n equations with Morley, discontinuous Galerkin, $C^0$ 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.

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arXiv2022

Numerical analysis of optimal control problems governed by fourth-order linear elliptic equations using the Hessian discretisation method

This paper focusses on the optimal control problems governed by fourth-order linear elliptic equations with clamped boundary conditions in the framework of the Hessian discretisation method (HDM). The HDM is an abstract framework that enables the convergence analysis of numerical methods through a quadruplet known as a Hessian discretisation (HD) and three core properties of HD. The HDM covers several numerical schemes such as the conforming finite element methods, the Adini and Morley non-conforming finite element methods (ncFEMs), method based on gradient recovery (GR) operators and the finite volume methods (FVMs). Basic error estimates and superconvergence results are established for the state, adjoint and control variables in the HDM framework. The article concludes with numerical results that illustrates the theoretical convergence rates for the GR method, Adini ncFEM and FVM.

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arXiv2021

A posteriori error analysis for a distributed optimal control problem governed by the von Kármán equations

This article discusses numerical analysis of the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain in $\mathbb{R}^2$. 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.

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arXiv2021

Conforming and Nonconforming Finite Element Methods for Biharmonic Inverse Source Problem

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 regularisation is considered to obtain a stable approximate solution. Error estimate is established for the regularised solution for different regularisation schemes. Numerical results that confirm the theoretical results are also presented.

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arXiv2020

Morley Finite Element Method for the von Kármán Obstacle Problem

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.

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arXiv2020

Hessian discretisation method for fourth order semi-linear elliptic equations: applications to the von Kármán and Navier--Stokes models

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 Kármán equations of plate bending are discussed. Results of numerical experiments are presented for the Morley ncFEM and GR method.

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arXiv2020

Improved $L^2$ and $H^1$ error estimates for the Hessian discretisation method

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.

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arXiv2018

The Hessian discretisation method for fourth order linear elliptic equations

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 $\mathbb{P}_1$ finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme and for a finite volume scheme.

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arXiv2017

Numerical analysis for the pure Neumann control problem using the gradient discretisation method

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, nonconforming and mimetic finite difference methods confirm the theoretical rates of convergence.

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arXiv2016

The gradient discretisation method for optimal control problems, with super-convergence for non-conforming finite elements and mixed-hybrid mimetic finite differences

In this paper, optimal control problems governed by diffusion equations with Dirichlet and Neumann boundary conditions are investigated in the framework of the gradient discretisation 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 is discussed. These super-convergence results are shown to apply to non-conforming $\mathbb{P}_1$ 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.

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