MB1: Recent Developments in Finite Difference and Finite Element Methods II

Room: Old Main Academic Center 3030

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OrganizerAmanda E Diegel, Mississippi State University

Davood Damircheli, Mississippi State University

Time: 10:00 am - 10:25 am (CST)

Title: On a Numerical Solution and Sensitivity Analysis of Timoshenko Beam with Discontinuous Petrov Galerkin Method

Abstract: In this investigation, we propose a new Ultraweak Discontinuous Petrov-Galerkin method to numerically analyze a one-dimensional Reissmer-Mindlin plate model which is a Timoshenko beam model. The governing equation arising from this problem is a fourth-order partial differential equation, and the numerical method proposed to approximate its solution is automatically stable and lock-free. Besides, a path-dependent sensitivity analysis of the Timoshenko problem will be performed for the first time. Numerical solutions illustrate the efficiency and convergence of the method especially when the thickness of the beam tends to zero.

Max Xue, University of North Carolina at Greensboro

Time: 10:25 am - 10:50 am (CST)

Title: Convergence Analysis of Non-Monotone Finite Difference Methods for Approximating Viscosity Solutions of Stationary Hamilton-Jacobi Equations

Abstract: A new non-monotone finite difference (FD) approximation method is proposed for stationary Hamilton-Jacobi problems with Dirichlet boundary condition. The FD method has local truncation errors that are above the first order Godunov barrier for monotone methods, and it is proved to converge to the unique viscosity solution of the underlying first order fully nonlinear partial differential equation. A stabilization term called a numerical moment is used to ensure the proposed scheme is admissible, stable, and convergent. Numerical tests are provided that compare the accuracy of the proposed scheme to that of the Lax-Friedrich’s method.

Joint work with Tom Lewis.

Satyajith Bommana Boyana, University of North Carolina at Greensboro

Time: 10:50 am - 11:15 am (CST)

Title: Symmetric Dual-Wind Discontinuous Galerkin Methods for a Parabolic Obstacle Problem

Abstract: Many applications such as phase transition problems, elasto plastic material behavior, option pricing, etc. deal with parabolic obstacle problems. The theoretical and the numerical analysis of such obstacle problems is challenging since the problem is nonlinear due to the presence of the obstacle function.

In this research project, we proposed and studied a fully discrete scheme to solve the parabolic variational inequality with a general obstacle function in ℝ2 that uses a symmetric dual-wind discontinuous Galerkin discretization in space and a backward Euler discretization in time. We established the convergence of numerical solutions in L(L2) and L2(H1) like energy norms and computed the rates. Numerical results are provided to demonstrate the performance of the proposed methods.

Joint work with Tom Lewis, Aaron Rapp and Yi Zhang.

Siamak Ghorbani Faal, Worcester Polytechnic Institute

Time: 11:15 am - 11:40 am (CST)

Title: Robust BPX Solver for Cahn-Hilliard Equations

Abstract: We present a choice of schemes to obtain stable and robust Newton’s iterations when solving the Cahn-Hilliard Equations with a logarithmic nonlinear potential function. Our approach utilizes splitting of the potential function into a sum of convex and concave parts that are, respectively, treated implicitly and explicitly. We propose a BPX type preconditioner for the finite element approximation of the problem that is robust with respect to spacial mesh and time step size.

Joint work with Adam Powell and Marcus Sarkis.

Marcus Sarkis-Martins, Worcester Polytechnic Institute

Time: 11:40 am - 12:05 pm (CST)

Title: Robust Model Reductions for Elliptic Problems with Heterogeneous High-Contrast Coefficients

Abstract: The goal of this talk is to present finite element discretizations for second-order elliptic problems with heterogeneous possibly high-contrast coefficients. Basedon a class of adaptive domain decomposition preconditioners named Balancing Domain Decomposition with Constraints-BDDC, Variational Multiscale Methods-VMS and Localized Orthogonal Decomposition Methods-LOD, we design robust model reductions given an energy target error.

Joint work with Alexandre Madureira from LNCC Brazil.