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[CT114]

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

  • Session Date & Time : 3E (Aug.23, 17:40-19:20)
  • 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
  • Author(s) :
    • Janitha Gunatilake (University of Peradeniya)

[00522] Modeling of concentration and electric field dependent susceptibilities in electrolytes

  • Session Date & Time : 3E (Aug.23, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : Electrolytes are everywhere: in fundamental electrochemistry, biochemical-systems, semiconductors, and many industrial devices. Their mathematical description is mainly characterised by Poisson-Nernst-Planck-type equations and their successors. However, the dielectric susceptibility of the Poisson equation is frequently considered as a constant. In this talk we derive a thermodynamically consistent concentration and electric field dependent susceptibility. We provide insights in the resulting equation system, discuss some important theoretical aspects and show its impact on the electrochemical double layer.
  • Classification : 65N30, 78A57, 80A17, 35Q61
  • Author(s) :
    • Manuel Landstorfer (Weierstrass Institute for Applied Analysis and Stochastics (WIAS)Weierstrass Institute for Applied Analysis and Stochastics (WIAS))
    • Jürgen Fuhrmann (Weierstrass Institute for Applied Analysis and Stochastics (WIAS)Weierstrass Institute for Applied Analysis and Stochastics (WIAS))
    • Rüdiger Müller (Weierstrass Institute for Applied Analysis and Stochastics (WIAS)Weierstrass Institute for Applied Analysis and Stochastics (WIAS))

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

  • Session Date & Time : 3E (Aug.23, 17:40-19:20)
  • 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
  • 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 Date & Time : 3E (Aug.23, 17:40-19:20)
  • 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
  • Author(s) :
    • Ritesh Ritesh (Indian Institute of Technology, Delhi)
    • Rohit Khandelwal (Indian Institute of Technology, Delhi)
    • Kamana Porwal (Indian Institute of Technology, Delhi)

[00799] Local Exactness of de Rham Conforming Hierarchical B-spline Differential Forms

  • Session Date & Time : 3E (Aug.23, 17:40-19:20)
  • Type : Contributed Talk
  • Abstract : Conservation laws present in partial differential equations arising in fluid mechanics and electromagnetics are frequently described using the de Rham sequence of differential forms. Stability of numerical methods solving these equations requires discrete preservation of these conservation laws. This talk will present sufficient local exactness criteria for a set of smooth, high-order, isogeometric, locally-refinable spline spaces in Euclidean space of arbitrary dimension in order to enable stable high-order, geometrically-precise finite element analyses.
  • Classification : 65N30, 58A12
  • Author(s) :
    • Kendrick M Shepherd (Brigham Young University)
    • Deepesh Toshniwal (TU Delft)

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

  • Session Date & Time : 4C (Aug.24, 13:20-15:00)
  • 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
  • Author(s) :
    • Tanvi Tanvi (Research Scholar)
    • Kamana Porwal (IIT DELHI)

[00899] Unlocking the Secrets of Locking

  • Session Date & Time : 4C (Aug.24, 13:20-15:00)
  • 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
  • Author(s) :
    • Charles Parker (University of Oxford)
    • Mark Ainsworth (Brown University)

[01104] Generation of $hp$-FEM Massive Databases for Deep Learning Inversion

  • Session Date & Time : 4C (Aug.24, 13:20-15:00)
  • Type : Contributed Talk
  • Abstract : Deep Neural Networks are employed in many geophysical applications to characterize the Earth’s subsurface. However, they often need to solve hundreds of thousands of complex and expensive forward problems to produce the training dataset. This work presents a robust approach to producing massive databases at a reduced computational cost. In particular, we build a single $hp$-adapted mesh that accurately solves many FEM problems for any combination of parameters within a given range.
  • Classification : 65N30, Finite Element Method, Deep Neural Networks, Goal-Oriented Adaptivity
  • Author(s) :
    • Julen Alvarez-Aramberri (University of the Basque Country (UPV/EHU))
    • Vincent Darrigrand (CNRS-IRIT, Toulouse)
    • Felipe Vinicio Caro (Basque Center for Applied Mathematics (BCAM), University of the Basque Country (UPV/EHU))
    • David Pardo (University of the Basque Country (UPV-EHU), Basque Center for Applied Mathematics (BCAM), Ikerbasque)

[01113] Acoustic Scattering Highly Efficient Iterative Method with high Order ABC

  • Session Date & Time : 4C (Aug.24, 13:20-15:00)
  • Type : Contributed Talk
  • Abstract : In this paper, we have developed a highly efficient numerical method for acoustic multiple scattering. This novel method consists of a high order local absorbing boundary condition combined with an isogeometric finite element and finite differences methods. By employing high order NURB basis, a globally high order method results. In our numerical experiments, we obtain errors close to machine precision by appropriate implementation of p- and h-refinement. We include numerical results which demonstrate the improved accuracy and efficiency of this new formulation compared with similar methods.
  • Classification : 65N30, 65N06, 65N85
  • Author(s) :
    • Vianey Roman Villamizar (Brigham Young University)
    • Vianey Roman Villamizar (Brigham Young UniversityBrigham Young University)
    • Tahsin Khajah (University of Texas at Tyler)
    • Jonathan Hale (University of Wisconsin at Madison)

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

  • Session Date & Time : 4C (Aug.24, 13:20-15:00)
  • 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
  • Author(s) :
    • SANGITA DEY (Ph.D Student of Indian Institute of Technology Kanpur)
    • Rathish Kumar Venkatesulu Bayya (Indian Institute of Technology Kanpur)

[02112] Adaptive Virtual Element Methods: convergence and optimality

  • Session Date & Time : 4D (Aug.24, 15:30-17:10)
  • 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
  • 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)

[02202] Deep Petrov-Galerkin Method for Solving Partial Differential Equations

  • Session Date & Time : 4D (Aug.24, 15:30-17:10)
  • Type : Contributed Talk
  • Abstract : In this talk, we introduce a new approach, Deep Petrov-Galerkin Method (DPGM), to solve partial differential equations (PDEs). This method is based on randomized neural networks for approximating the unknown functions and allows for a flexible selection of test functions, including finite element basis functions or other neural networks. Although the resulting linear system may not be symmetric or square, it can still be solved through a least-squares method. We demonstrate the effectiveness of our approach through a variety of use cases, including mixed DPGMs for solving Poisson problems and two classical time-dependent problems using a space-time approach, where the temporal and spatial variables are treated equally and the initial conditions serve as boundary conditions. Additionally, we use the Burger's equation to illustrate how DPGM can be applied to non-linear PDEs. Our numerical results show the superiority of DPGM over traditional methods such as finite element and finite difference methods. The advantages of DPGM include improved accuracy with reduced degrees of freedom, ease of implementation due to its mesh-free nature, good flexibility in handling different boundary conditions, and efficient solving time-dependent problems. These results highlight the strong potential of DPGM in the field of numerical methods for PDEs.
  • Classification : 65N30, 68T07
  • Author(s) :
    • Yong Shang (Xi'an Jiaotong University)
    • Fei Wang (Xi'an Jiaotong University)
    • Jingbo Sun (Xi'an Jiaotong University)

[02496] Convergence of Morley Adaptive FEM for Distributed Optimal Control Problems

  • Session Date & Time : 4D (Aug.24, 15:30-17:10)
  • Type : Contributed Talk
  • Abstract : This talk will discuss the quasi-optimality of adaptive nonconforming finite element methods (FEM) for a distributed optimal control problem governed by the biharmonic operator. Morley finite element spaces are used for discretizing the state and adjoint variables over a triangular mesh, and a variational discretization approach is used for control. The adaptive FEM's quasi-optimality stems from its general axiomatic framework, which includes stability, reduction, discrete reliability, and quasi-orthogonality. Numerical results confirm the theoretical convergence orders.
  • Classification : 65N30, 65N15, Adaptive finite element methods, optimal control problems, Non-conforming FEM
  • Author(s) :
    • Asha Kisan Dond (Indian Institute of Science Education and Research Thiruvananthapuram)
    • Neela Nataraj (Indian Institute of Technology Bombay)
    • Subham Nayak (Indian Institute of Science Education and Research Thiruvananthapuram)

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

  • Session Date & Time : 4D (Aug.24, 15:30-17:10)
  • 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
  • Author(s) :
    • Jones Tarcius Doss (Department of Mathematics, Anna University, Chennai)

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

  • Session Date & Time : 4D (Aug.24, 15:30-17:10)
  • 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
  • Author(s) :
    • Aaron Matthew Baier-Reinio (University of Oxford)
    • Patrick Farrell (University of Oxford)