Nonconforming virtual element method for an incompressible miscible displacement problem in porous media Sarvesh Kumar, Devika Shylaja Computers and Mathematics with Applications, 2025 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 ( div ) conforming virtual element method 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.
Convergence Analysis of a Nonconforming Virtual Element Method for Compressible Miscible Displacement Problems in Porous Media Sarvesh Kumar, Devika Shylaja Numerical Methods for Partial Differential Equations, 2025 This article presents a priori error estimates for the miscible displacement of one compressible fluid by another in a porous medium. The study utilizes the 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 discretized 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.
Numerical analysis of optimal control problems governed by fourth-order linear elliptic equations using the Hessian discretization method Devika Shylaja Optimal Control Applications and Methods, 2024 This article focuses on the optimal control problems governed by fourth‐order linear elliptic equations with clamped boundary conditions in the framework of the Hessian discretization method (HDM). The HDM is an abstract framework that enables the convergence analysis of several numerical methods such as the conforming finite element methods, the Adini and Morley nonconforming finite element methods (ncFEMs), the 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. A companion operator for the GR method with specific property is designed. The article concludes with numerical results that illustrate the theoretical convergence rates for GR method, Adini ncFEM, and FVM.
Unified a priori analysis of four second-order FEM for fourth-order quadratic semilinear problems Carsten Carstensen, Neela Nataraj, Gopikrishnan C. Remesan, Devika Shylaja Numerische Mathematik, 2023 A 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.
A posteriori error analysis for a distributed optimal control problem governed by the von Kármán equations Sudipto Chowdhury, Asha K. Dond, Neela Nataraj, Devika Shylaja ESAIM Mathematical Modelling and Numerical Analysis, 2022 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.
Conforming and nonconforming finite element methods for biharmonic inverse source problem M Thamban Nair, Devika Shylaja Inverse Problems, 2022 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.
Morley finite element method for the von Kármán obstacle problem Carsten Carstensen, Sharat Gaddam, Neela Nataraj, Amiya K. Pani, Devika Shylaja ESAIM Mathematical Modelling and Numerical Analysis, 2021 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.
Hessian discretisation method for fourth-order semi-linear elliptic equations: applications to the von Kármán and Navier–Stokes models Jérome Droniou, Neela Nataraj, Devika Shylaja Advances in Computational Mathematics, 2021 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.
Morley FEM for a distributed optimal control problem governed by the von Kármán equations Sudipto Chowdhury, Neela Nataraj, Devika Shylaja Computational Methods in Applied Mathematics, 2021 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.
