MA1: Recent Developments in Finite Difference and Finite Element Methods I

Room: Old Main Academic Center 3030

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Organizer: Amanda E Diegel, Mississippi State University

Tom Lewis, University of North Carolina at Greensboro

Time: 10:00 am - 10:25 am (CST)

Title: Narrow-stencil Numerical Approximations of Fully Nonlinear Second Order PDEs

Abstract: We will present a new general framework for designing convergent finite difference methods for approximating viscosity and regular solutions of fully nonlinear second order partial differential equations. Unlike the well-known monotone framework, the proposed new framework allows for the use of narrow stencils. The main building blocks for the new narrow-stencil framework are various one-sided first order and various symmetric second order numerical derivative operators which are then used to define appropriate notions of consistency and g-monotonicity. Specific methods that satisfy the framework are constructed using numerical moments, and analytic properties such as admissibility, stability, and convergence will be discussed. Numerical experiments will be provided, and a generalization for discontinuous Galerkin methods that allow for high order and unstructured meshes will be introduced.

Natasha Sharma, University of Texas at El Paso 

Time: 10:25 am - 10:50 am (CST)

Title: Unconditional Energy Stability and Solvability for a C0 Interior Penalty Method for a Sixth-Order Equation Modeling Microemulsions

Abstract: In this talk, we present a continuous interior penalty Galerkin method for solving a certain class of sixth-order Cahn-Hillard type equation which models the dynamics of phase transitions in ternary oil-water-surfactant systems. We express this nonlinear sixth-order parabolic equation in its mixed form as a system consisting of a second-order (in space) parabolic equation and an algebraic fourth-order (in space) nonlinear equation. We choose a time discretization so that a discrete energy law can be established leading to unconditional energy stability. Furthermore, we show that the numerical method is unconditionally uniquely solvable. We close the talk by presenting the numerical results of some benchmark problems to verify the practical performance of the proposed approach and discuss some exciting current and future applications including adaptive time-stepping strategies which seek to resolve the different time scales that arise in the interfacial dynamics of the system.

Joint work with Amanda E Diegel.

Aaron Rapp, University of the Virginia Islands

Time: 10:50 am - 11:15 am (CST)

Title: Consistency and Penalty Parameter Results for the Dual-Wind Discontinuous Galerkin Methods

Abstract: A discontinuous Galerkin (DG) finite-element interior calculus is used as a common framework to describe various DG approximation methods for second-order elliptic problems. This framework allows for the approximation of both primal and variational forms of second order differential equations. In this presentation, we will study the error from using the dual-wind DG derivatives to approximate the Laplacian for the primal form of the Poisson equation. We will also study the convergence of our discontinuous Galerkin space to a continuous Galerkin space. Some analytical results will be presented, along with numerical examples that verify these results.

José Garay, Louisiana State University 

Time: 11:15 am - 11:40 am (CST)

Title: Multiscale Finite Element Methods For An Elliptic Optimal Control Problem With Rough Coefficients

Abstract: The solution of multiscale elliptic problems with non-separable scales and high contrast in the coefficients by standard Finite Element Methods (FEM) is typically prohibitively expensive since it requires the resolution of all characteristic lengths to produce an accurate solution. Numerical homogenization methods such as Localized Orthogonal Decomposition (LOD) methods provide access to feasible and reliable simulations of such multiscale problems. These methods are based on the idea of a generalized finite element space where the generalized basis functions are obtained by modifying standard coarse FEM basis functions to incorporate relevant microscopic information in a computationally feasible procedure. Using this enhanced basis, one can solve a much smaller problem to produce an approximate solution whose accuracy is comparable to the solution obtained by the expensive standard FEM. Based on the LOD methodology, we investigate multiscale finite element methods for an elliptic distributed optimal control problem with rough coefficients. Numerical results obtained with our parallel implementation of the method illustrate our theoretical results.

Joint work with Susanne C Brenner and Li-yeng Sung.

Zhiyu Tan, Louisiana State University

Time: 11:40 am - 12:05 pm (CST)

TitleA Convexity Enforcing C0 Interior Penalty Method for the Monge-Ampère Equation on Strictly Convex Smooth Planar Domains

Abstract: We extend the work in [S. C. Brenner et al., Numer. Math.,148(2021): 497-524], where a C0 interior penalty method for the Dirichlet boundary value problem of the Monge-Ampère equation on convex polygonal domains is proposed and analyzed, to strictly convex smooth planar domains. A new isoparametric finite element based on a new enhanced modified cubic Lagrange finite element and a cubic isoparametric map is introduced. The C0 interior penalty method is based on the enhanced cubic Lagrange finite element and this isoparametric finite element, which together enable the enforcement of the convexity of the approximation solutions. A priori and a posteriori error estimates are obtained. Numerical results are also presented to corroborate the theoretical results.