# Registered Data

Contents

- 1 [CT114]
- 1.1 [00175] pFemView: An Open-Source Visualization Library for p-FEM
- 1.2 [00577] Mixed Finite Element Method for Dirichlet Boundary Optimal Control Problem
- 1.3 [00741] POINTWISE ADAPTIVE QUADRATIC DG FEM FOR OBSTACLE PROBLEM
- 1.4 [01573] A novel robust adaptive algorithm for time fractional diffusion wave equation on non-uniform meshes
- 1.5 [01846] L3 approximation of the Caputo derivatives and its application to time-fractional wave equation
- 1.6 [00899] Unlocking the Secrets of Locking
- 1.7 [00611] Deformations of linear elastic bodies computed using the RBF-PU method
- 1.8 [01118] Finite Element Analysis of a Non-equilibrium Model for Hybrid Nano-Fluid
- 1.9 [00821] ADAPTIVE QUADRATIC DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR THE UNILATERAL CONTACT PROBLEM
- 1.10 [02603] AN $H^1$ GALERKIN MIXED FINITE ELEMENT METHOD FOR ROSENAU EQUATION
- 1.11 [02112] Adaptive Virtual Element Methods: convergence and optimality
- 1.12 [02640] High-Order Finite Element Schemes for Multicomponent Flow Problems
- 1.13 [02174] A mathematical model to predict how obesity raises the risk of diabetes
- 1.14 [01815] Time-fractional SVIR chicken-pox mathematical model with quarantine compartment

# [CT114]

**Session Time & Room**- CT114 (1/3) :
__3E__@__E711__[Chair: Janitha Gunatilake] - CT114 (2/3) :
__4C__@__E711__[Chair: Charles Parker] - CT114 (3/3) :
__4D__@__E711__[Chair: Jones Tarcius Doss] **Classification**- CT114 (1/3) : Numerical methods for partial differential equations, boundary value problems (
__65N__) - CT114 (2/3) : Numerical methods for partial differential equations, boundary value problems (
__65N__) - CT114 (3/3) : Numerical methods for partial differential equations, boundary value problems (
__65N__) / Mathematical biology in general (__92B__)

## [00175] pFemView: An Open-Source Visualization Library for p-FEM

**Session Time & Room**:__3E__(Aug.23, 17:40-19:20) @__E711__**Type**: Contributed Talk**Abstract**: We present a new approach to visualize p-hierarchical basis finite element (p-FEM) solutions on the scientific visualization application ParaView. Since ParaView uses a linear/quadratic interpolation at specific geometric nodes, a refined visualization mesh needs to be constructed efficiently. This is accomplished via the key steps “p-hierarchical to nodal projection” and “higher-order to lower-order projection”, which have been implemented in an open-source C++ library “pFemView”. Furthermore, examples are presented to illustrate the effectiveness of this library.**Classification**:__65N30__,__65Y15__,__68-04__**Format**: Talk at Waseda University**Author(s)**:**Janitha Gunatilake**(University of Peradeniya)

## [00577] Mixed Finite Element Method for Dirichlet Boundary Optimal Control Problem

**Session Time & Room**:__3E__(Aug.23, 17:40-19:20) @__E711__**Type**: Contributed Talk**Abstract**: The optimal control problems (OCPs) subjected to partial differential equations (PDEs) have numerous applications in fluid dynamics, image processing, mathematical finance etc. The objective of OCPs is to find the optimal control which minimizes or maximizes the given cost functional with certain constraints being satisfied. There are mainly two types of OCPs available in literature namely, Distributed Control Problems where the control acts on the system through an external force and Boundary Control Problems where the control acts on the system through a Dirichlet or Neumann or Robin boundary conditions. Dirichlet boundary control problems are difficult to handle due to variational difficulty. In many applications, it is important to obtain accurate approximation of the scalar variable and its gradient simultaneously. A common way to achieve this goal is to use mixed finite element methods. The main aim of my talk is to analyze the mixed finite element method for the second order Dirichlet boundary control problem in which the control is penalized in the energy space. Mixed finite element methods have the property that they maintain the discrete conservation law at the element level. For the variational formulation, the state equation is converted to the mixed system using the mixed variational scheme for second order elliptic equations and then the continuous optimality system is derived. In order to discretize the continuous optimality system, the lowest order Raviart-Thomas space is used to numerically approximate the state and co-state variables whereas the continuous piece-wise linear finite element space is used for the discretization of control. Based on this formulation, the optimal order a priori error estimates for the control in the energy norm and $L_2$-norm is derived. The reliability and the efficiency of proposed a posteriori error estimator is also discussed using the Helmholtz decomposition. Finally, several numerical experiments are presented to confirm the theoretical findings.**Classification**:__65N30__,__65N15__,__65N12__,__65K10__**Format**: Talk at Waseda University**Author(s)**:**Divay Garg**(Indian Institute of Technology Delhi)- Kamana Porwal (Indian Institute of Technology Delhi)

## [00741] POINTWISE ADAPTIVE QUADRATIC DG FEM FOR OBSTACLE PROBLEM

**Session Time & Room**:__3E__(Aug.23, 17:40-19:20) @__E711__**Type**: Contributed Talk**Abstract**: The obstacle problem, often considered as a prototype for a class of free boundary problems. The elliptic obstacle problem is a nonlinear model that describes the vertical movement of a object restricted to lie above a barrier $\text{(obstacle)}$ while subjected to a vertical force. In this talk, we perform a posteriori error analysis in the supremum norm for the quadratic Discontinuous Galerkin$\text{(DG)}$ method for the elliptic obstacle problem. Compare with the energy norm estimates, supremum norm estimates gives the pointwise control on the error. We have carried out the analysis on two different discrete sets, one set having integral constraints and other one with the nodal constraints at the quadrature points, and discuss the pointwise reliability and efficiency of the proposed a posteriori error estimator. In the analysis, we employ a linear averaging function to transfer DG finite element space to standard conforming finite element space and exploit the sharp bounds on the Green's function of the Poisson's problem. Moreover, the upper and the lower barrier functions corresponding to continuous solution $u$ are constructed by modifying the conforming part of the discrete solution $u_h$ appropriately. Finally, the numerical results for adaptive FEM are presented in order to exhibit the reliability and the efficiency of the proposed error estimator.**Classification**:__65N30__,__65N15__**Format**: Talk at Waseda University**Author(s)**:**Ritesh Ritesh**(Indian Institute of Technology, Delhi)- Rohit Khandelwal (Indian Institute of Technology, Delhi)
- Kamana Porwal (Indian Institute of Technology, Delhi)

## [01573] A novel robust adaptive algorithm for time fractional diffusion wave equation on non-uniform meshes

**Session Time & Room**:__3E__(Aug.23, 17:40-19:20) @__E711__**Type**: Contributed Talk**Abstract**: In this work, a novel high-order adaptive algorithm on non-uniform grid points for the Caputo fractional derivative is derived. Developed algorithm allows one to build adaptive nature where numerical scheme is adjusted according to behavior of $\alpha$ to keep errors very small and converge to solution very fast. Analysis of numerical scheme has been established thoroughly. Moreover, a reduced order technique is implemented by using moving mesh refinement to improve accuracy at several time levels.**Classification**:__65N06__,__65N12__,__65N50__,__65N15__**Format**: Online Talk on Zoom**Author(s)**:**Rahul Kumar Maurya**(Government Tilak P.G. College, Katni, Madhya Pradesh, India)- Vineet Kumar Singh (Indian Institute of Technology (BHU), Varanasi, India)

## [01846] L3 approximation of the Caputo derivatives and its application to time-fractional wave equation

**Session Time & Room**:__3E__(Aug.23, 17:40-19:20) @__E711__**Type**: Contributed Talk**Abstract**: In this talk, we will discuss a new second-order L3 approximation of the Caputo fractional derivative of order 1< α < 2. We have applied Lagrange’s cubic interpolating polynomial to develop this approximation. A second-order difference scheme is also proposed to find the numerical solution of the time-fractional wave equation. The numerical analysis results of the proposed algorithm are provided and a comparative study is also given to show the effectiveness and accuracy of the proposed scheme.**Classification**:__65N06__**Format**: Online Talk on Zoom**Author(s)**:**NIKHIL SRIVASTAVA**(Indian Institute of Technology (BHU) Varanasi, India)

## [00899] Unlocking the Secrets of Locking

**Session Time & Room**:__4C__(Aug.24, 13:20-15:00) @__E711__**Type**: Contributed Talk**Abstract**: For nearly incompressible linear elastic materials, such as rubber, finite element methods sometimes exhibit suboptimal convergence rates for the energy and/or stresses. This type of behavior, termed “locking”, is still not completely understood. This talk reviews the concept of locking and recent results that show that conforming high order finite elements provide optimal convergence for both the energy and stresses with respect to the mesh size and polynomial degree. Robust preconditioners will also be presented.**Classification**:__65N30__,__74S05__,__65N35__**Format**: Talk at Waseda University**Author(s)**:**Charles Parker**(University of Oxford)- Mark Ainsworth (Brown University)

## [00611] Deformations of linear elastic bodies computed using the RBF-PU method

**Session Time & Room**:__4C__(Aug.24, 13:20-15:00) @__E711__**Type**: Contributed Talk**Abstract**: In this talk we will present numerical solutions to boundary value problems of linear elasticity, computed using the Radial Basis Function Partition of Unity (RBF-PU) method in the least squares formulation. Specifically, we will show deformations of 3D geometries, including a thin plank under bending and a reconstructed human diaphragm under ventilation conditions. Convergence studies and a comparison of the RBF-PU method to the standard Galerkin Finite Element method (GFEM) will be presented.**Classification**:__65N35__,__74S25__,__74B05__**Format**: Talk at Waseda University**Author(s)**:**Andreas Michael**(Uppsala University)

## [01118] Finite Element Analysis of a Non-equilibrium Model for Hybrid Nano-Fluid

**Session Time & Room**:__4C__(Aug.24, 13:20-15:00) @__E711__**Type**: Contributed Talk**Abstract**: A theoretical and computational finite element study of modified Navier-Stokes Equations coupled with energy conservation governing the flow and heat transfer in complex domain with hybrid nanofluid is carried out. The apriori error estimates providing the convergence analysis for the finite element scheme is derived in the H1-norm. The effect of hybrid nano-particle’s volume fraction, Rayleigh Number, Prandtl Number, Darcy number, porosity are analyzed to trace the physics related to flow and heat transfer.**Classification**:__65N30__,__80A05__,__76R99__,__Finite element analysis and numerical computations with its application to hybrid nano-fluid__**Format**: Online Talk on Zoom**Author(s)**:**SANGITA DEY**(Ph.D Student of Indian Institute of Technology Kanpur)- Rathish Kumar Venkatesulu Bayya (Indian Institute of Technology Kanpur)

## [00821] ADAPTIVE QUADRATIC DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR THE UNILATERAL CONTACT PROBLEM

**Session Time & Room**:__4C__(Aug.24, 13:20-15:00) @__E711__**Type**: Contributed Talk**Abstract**: The proposed title of my talk will be ADAPTIVE QUADRATIC DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR THE UNILATERAL CONTACT PROBLEM . In the talk, I will be discussing about employing discontinuous Galerkin methods (DG) for the finite element approximation of frictionless unilateral contact problem using quadratic finite elements over simplicial triangulation. We shall analyze a posteriori error estimates in the DG norm wherein, the reliability and efficiency of the proposed a posteriori error estimators will be addressed. Further we will show that numerical results substantiate the theoretical findings,**Classification**:__65N30__,__65N15__**Format**: Talk at Waseda University**Author(s)**:**Tanvi Tanvi**(Research Scholar)- Kamana Porwal (IIT DELHI)

## [02603] AN $H^1$ GALERKIN MIXED FINITE ELEMENT METHOD FOR ROSENAU EQUATION

**Session Time & Room**:__4D__(Aug.24, 15:30-17:10) @__E711__**Type**: Contributed Talk**Abstract**: In this paper, by applying a splitting technique, the non-linear fourth order Rosenau equation is split into a system of coupled equations. Then, an $H^1$ Galerkin mixed finite element method is proposed for the resultant equations after employing a suitable weak formulation. Semi-discrete and fully discrete schemes are discussed and respective optimal order error estimates are obtained without any constraints on the mesh. Finally, numerical results are computed to validate the efficacy of the method. The proposed method has advantages in respect of higher order error estimate, less requirement of regularity on exact solution and also with reduced size i.e. less than half of the size of resulting linear system over that of mentioned in Manickam et al., Numerical Methods for Partial Differential Equations, (14), (1998), pp. 695-716.**Classification**:__65N30__,__65N06__,__65M60__,__65M06__**Format**: Talk at Waseda University**Author(s)**:**Jones Tarcius Doss**(Department of Mathematics, Anna University, Chennai)

## [02112] Adaptive Virtual Element Methods: convergence and optimality

**Session Time & Room**:__4D__(Aug.24, 15:30-17:10) @__E711__**Type**: Contributed Talk**Abstract**: We consider a Virtual Element discretization of elliptic boundary-value problems, using triangular or quadrilateral meshes with hanging nodes of arbitrary, but fixed, maximal index. We design a two-stage adaptive algorithm, based on a stabilization-free a posteriori error estimator, which alternates data approximation and solution approximation with increasing accuracy. We prove the convergence of the inner and outer loops, we establish the optimality of the adaptive procedure in suitable approximation classes, and we provide numerical results.**Classification**:__65N30__,__65N50__**Format**: Talk at Waseda University**Author(s)**:**Claudio Canuto**(Politecnico di Torino)- Lourenco Beirao da Veiga (University of Milan Bicocca)
- Ricardo H Nochetto (University of Maryland)
- Giuseppe Vacca (University of Bari)
- Marco Verani (Politecnico di Milano)

## [02640] High-Order Finite Element Schemes for Multicomponent Flow Problems

**Session Time & Room**:__4D__(Aug.24, 15:30-17:10) @__E711__**Type**: Contributed Talk**Abstract**: The Stokes–Onsager–Stefan–Maxwell (SOSM) equations model the flow of concentrated mixtures of distinct chemical species in a common thermodynamic phase. We derive a novel variational formulation of these nonlinear equations in which the species mass fluxes are treated as unknowns. This new formulation leads to a large class of high-order finite element schemes with desirable linear-algebraic properties. The schemes are provably convergent when applied to a linearization of the SOSM problem.**Classification**:__65N30__,__76T30__,__35Q35__**Format**: Talk at Waseda University**Author(s)**:**Aaron Matthew Baier-Reinio**(University of Oxford)- Patrick Farrell (University of Oxford)

## [02174] A mathematical model to predict how obesity raises the risk of diabetes

**Session Time & Room**:__4D__(Aug.24, 15:30-17:10) @__E711__**Type**: Contributed Talk**Abstract**: Nowadays, obesity is a serious global issue. Obesity increases the risk of developing significant health issues like diabetes, cancer, and heart attacks. This work tries to depict the link between pancreatic damage, blood insulin levels, and blood glucose in a mathematical model. The model also illustrates how the increased obesity index raises diabetes risk. Additionally, we incorporated a delay term in the model to depict insulin production lag brought on by dysfunctional beta-cells due to obesity. We analytically analyzed both delay and non-delay models. Moreover, numerical simulations are demonstrated to support the theoretically-based analysis.**Classification**:__92B05__,__34H05__,__34D05__**Author(s)**:**Parimita Roy**(Thapar Institute of Engineering and Technology)- Ani Jain (Thapar Institute of Engineering and Technology)

## [01815] Time-fractional SVIR chicken-pox mathematical model with quarantine compartment

**Session Time & Room**:__4D__(Aug.24, 15:30-17:10) @__E711__**Type**: Contributed Talk**Abstract**: This work considers a time-fractional SVIR chicken-pox reaction-diffusion model with nonlinear diffusion operators. The model also contains the quarantine compartment and therefore, it consists of five unknown variables. Further suitable initial and boundary conditions are also given along with the model. The existence of weak solutions proved for the proposed time-fractional model in the bounded domain with appropriate assumptions and a-priori energy estimates. The main results of the work demonstrated using the Faedo-Galerkin method and approximation problem. Finally, numerical simulations are provided to understand the evolution of the chicken-pox virus among the population.**Classification**:__92B05__,__35K57__,__35A01__**Author(s)**:**Shangerganesh Lingeshwaran**(National Institute of Technology Goa)- Hariharan Soundararajan (National Institute of Technology Goa)
- Manimaran Jeyaraj (Vellore Insitute of Technology)