MiniSymposium-2: Recent Advances in Efficient Computation for Partial Differential Equations
Many large-scale problems are modelled by partial differential equations (PDEs) and can only be solved numerically. It is extremely important to improve the efficiency and accuracy of the numerical computations. The aim of this minisymposium is to exchange and discuss ideas for the recent advances in numerical solutions for PDEs. The focus will be on the improvement of the computational efficiency via various approaches including but not limited to mesh adaptation/moving mesh methods, adaptive discretization methods, adaptive finite element method, parallel computing, and their applications to solve science and engineering problems. It provides the researchers from different groups with the opportunity to discuss the related topics and relate their research to form possible collaborative projects.
Xianping Li, Arizona State University
Saturday, October 2, 2021 at 10:20 – 11:40 am (CST) (Session 1)
Arizona State University
Moving Mesh with Streamline Upwind Petrov-Galerkin (MM-SUPG) Method for Convection-Dominated Convection-Diffusion Problems
We investigate the effect of the streamline upwind Petrov-Galerkin method (SUPG) in relation to the moving mesh partial differential equation (MMPDE) method for convection-diffusion problems in the presence of vanishing diffusivity ε. We consider solutions that initial sharp layers on the interior of the domain and flow fields that propogate those layers for t ∈ (0,T]. On a fixed mesh, SUPG (FM-SUPG) is shown to enhance stability of spurious oscillations when compared to the classic Galerkin method (FM-FEM) when ε is small. However, for these types of problems, FM-SUPG falls short when the layer-gradient is large. We resolve this issue using a variational method for anisotropic mesh adaptation. Examples are provided that demonstrate the benefits of supplementing the variational mesh generation approach with SUPG, resulting in smaller errors in the H1(Ω) norm. Additionally, visual inspection demonstrates that MM-SUPG provides smoother solutions than FM-FEM and FM-SUPG.
Purdue University Northwest
Boundary Conditions for Constrained Hyperbolic Systems of Partial Differential Equations
Many mathematical models in science and technology are based on hyperbolic systems of differential equations whose solutions must satisfy certain constraints. When the models are restricted to bounded domains, the problem of well-posed, constraint-preserving boundary conditions arises naturally. However, for numerical solutions finding such boundary conditions may represent just a step in the right direction. Including the constraints as dynamical variables of a larger, unconstrained system associated to the original one could provide better numerical results, as the constraints are kept under control during evolution. One of the main goals of the talk is to present this idea in the case of constrained first order symmetric hyperbolic systems of differential equations subject to maximal nonnegative boundary conditions.
Wright State University
Domain Decomposition Moving Mesh Method for Solving Phase-Field Models
We present an efficient domain decomposition (DD) moving mesh method for solving phase-field modes in two and three spatial dimensions. The computational domain is partitioned into overlapping subdomains, and the adaptive mesh method is employed on each subdomain to compute the solution of the phase-field model. The global solution for the phase-field model is then reconstructed using a DD approach similar to the Schwarz alternating method. We present several numerical experiments to demonstrate the performance of the domain decomposition moving mesh method. The numerical results illustrate the efficiency of the proposed DD moving mesh method for solving phase-field models.
University of Houston
A Finite Element Method for Two-phase Surface Fluids and Modeling of Multicomponent Lipid Membranes
This talk reviews a continuum-based model for the process of phase separation in multicomponent lipid membranes exhibiting lateral fluidity. We further introduce a finite element method for solving surface fluid and surface phase-field equations. The models and methods are combined to deliver a finite element method for a thermodynamically consistent phase-field model for surface two-phase fluid. A stable linear splitting approach is introduced and available numerical analysis results are presented. We finally discuss successes and failures of the model to reproduce in vitro experiments with multicomponent vesicles of different lipid compositions.
Saturday, October 2, 2021 at 2:40 – 4:00 pm (CST) (Session 2)
Memorial University, NL, Canada
A Fully Discrete Analysis of Schwarz Waveform Relaxation for the Heat Equation
Schwarz waveform relaxation (SWR) is a domain decomposition method for the parallel solution of time dependent PDEs. The computational space-time domain is partitioned and the problem is iteratively solved over these subdomains. Previous analysis has been provided by Gander et al. in the continuous and semi-discrete (in space) cases. Here we provide an analysis of SWR in the fully discrete case and compare our results to the existing continuous and semi-discrete results.
Di (Richard) Liu
Michigan State University
Multiscale Modeling and Computation of Optically Manipulated Nano Devices
We present a multiscale modeling and computational scheme for optical-mechanical responses of nanostructures. The multi-physical nature of the problem is a result of the interaction between the electromagnetic (EM) field, the molecular motion, and the electronic excitation. To balance accuracy and complexity, we adopt the semi-classical approach that the EM field is described classically by the Maxwell equations, and the charged particles follow the Schrödnger equations quantum mechanically. To overcome the numerical challenge of solving the high dimensional multi-component many- body Schrödinger equations, we further simplify the model with the Ehrenfest molecular dynamics to determine the motion of the nuclei, and use the Time- Dependent Current Density Functional Theory (TD-CDFT) to calculate the excitation of the electrons. This leads to a system of coupled equations that computes the electromagnetic field, the nuclear positions, and the electronic current and charge densities simultaneously. In the regime of linear responses, the resonant frequencies initiating the out-of-equilibrium optical-mechanical responses can be formulated as an eigenvalue problem. A self-consistent multiscale method is designed to deal with the well separated space scales. The isomerization of Azobenzene is presented as a numerical example.
Texas A&M University-Corpus Christi
Mutliscale Methods, Upscaling and Machine Learning Techniques for Poroelasticity Problems in Heterogeneous Media
In this work, we consider poroelastcity problems in heterogeneous media. We present upscaling and multiscale methods for construction of a coarse-grid model. We propose a rigorous and accurate multiscale solver and upscaling framework based on some recently developed multiscale methods. In the proposed multiscale method, we identify macroscale parameters via appropriate local solution. The method involves two basic steps: (1) the construction of multiscale basic functions that take into account small scale heterogeneity in the local domains and (2) the macroscopic equations for the coarse-scale model. To accelerate construction of the coarse grid model, we apply machine learning technique for fast and accurate calculation of macroscopic parameters. We train neural networks on a set of selected realizations of local microscale stochastic fields and macroscale characteristics. We construct a deep learning method through a convolutional neural network (CNN) to learn a map between stochastic fields and macroscopic parameters. Numerical results are presented for two and three-dimensional model problems and show that the proposed method provides fast and accurate macroscale parameters predictions.
Washington University-St. Louis
Hybridization and Postprocessing in Finite Element Exterior Calculus
Finite element exterior calculus (FEEC) unifies several families of conforming finite element methods for Laplace-type problems, including the scalar and vector Poisson equations. This talk presents a framework for hybridization of FEEC, which recovers known hybrid methods for the scalar Poisson equation and gives new hybrid methods for the vector Poisson equation. We also generalize Stenberg postprocessing, proving new superconvergence estimates. Based on joint work with Gerard Awanou, Maurice Fabien, and Johnny Guzman.
Saturday, October 2, 2021 at 4:40 – 6:00 pm (CST) (Session 3)
Convergence of a Finite Element Method for Two-Phase Flows in Porous Media
Simulations of immiscible two-phase flows in heterogeneous porous media at the Darcy scale are important in the understanding of flow and transport processes in subsurface. Applications include storage of carbon dioxide in saline aquifers and production of hydrocarbons from oil and gas reservoirs. Mathematical models are based on conservation of mass for each fluid phase and they are characterized by systems of nonlinear coupled partial differential equations. In this talk, we formulate a finite element method for solving the wetting phase pressure and saturation of an incompressible immiscible two-phase system. Challenges in the numerical analysis arise from the nonlinearity and the unboundedness or degeneracy of some of the PDE coefficients. Recent advances on the theoretical convergence of the proposed schemes for the discretization of the mathematical model in the general case are presented. Accuracy and robustness of the numerical method for heterogeneous porous media are demonstrated.
Pennsylvania State University
A Posteriori Error Estimates in Finite Element Method by Preconditioning
We present a framework that relates preconditioning with a posteriori error estimates in finite element methods. In particular, we use standard tools in subspace correction methods to obtain reliable and efficient error estimators. As a simple example, we recover the classical residual error estimators for the second order elliptic equations as well as present some new estimators for systems of PDEs. This is joint work with Yuwen Li (Penn State).
University of North Carolina-Greensboro
Narrow-Stencil Approximation Methods for Fully Nonlinear Elliptic Boundary Value Problems
This talk will introduce a new convergent narrow-stencil finite difference method for approximating viscosity solutions of second order fully nonlinear elliptic partial differential equations including Hamilton-Jacobi-Bellman equations. The proposed finite difference method naturally extends the Lax-Friedrichs method for first order problems to second order problems by introducing a key stabilization term called a numerical moment. By abandoning the standard monotonicity assumption, the new methods do not require the use of wide-stencils. The new narrow-stencil methods are easy to formulate and implement, and they formally have higher-order truncation errors than monotone methods when first-order terms are present in the PDE. We will also discuss a new discontinuous Galerkin method that formally extends the narrow-stencil finite difference methods to achieve higher order accuracy.
University of Pittsburgh
A Divergence-free Finite Element Method for the Stokes Problem with Boundary Correction
In this talk we discuss a boundary correction finite element method for the Stokes problem based on the Scott-Vogelius pair on Clough-Tocher splits. The velocity space consists of continuous piecewise quadratic polynomials, and the pressure space consists of piecewise linear polynomials without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise quadratic polynomials with respect to boundary partition is introduced to enforce (normal) boundary conditions and to mitigate the lack of pressure-robustness. We prove that the method converges with optimal order and the velocity approximation is divergence free. This is joint work with Baris Otus and Haoran Liu (Pitt).
Sunday, October 3, 2021 at 11:00 am – 12:20 pm (CST) (Session 4)
University of Illinois-Chicago
The Second Boundary Value Problem for a Discrete Monge-Ampere Equation
In this work we propose a natural discretization of the second boundary condition for the Monge-Ampere equation of geometric optics and optimal transport. It is the natural generalization of the popular Oliker-Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence, unicity and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.
University of Delaware
Boundary Control of Time-harmonic Eddy Current Equations
Motivated by various applications, this article develops the notion of boundary control for Maxwell’s equations in the frequency domain. Surface curl is shown to be the appropriate regularization in order for the optimal control problem to be well-posed. Since all underlying variables are assumed to be complex valued, the standard results on differentiability do not directly apply. Instead, we extend the notion of Wirtinger derivatives to complexified Hilbert spaces. Optimality conditions are rigorously derived and regularity of the adjoint variable is established. The state and adjoint variables are discretized using higher-order Nédélec finite elements. The finite element space for controls is identified, as a space, which preserves the structure of the control regularization. Convergence of the fully discrete scheme is established. The theory is validated by numerical experiments, in some cases, motivated by realistic applications.
University of Connecticut
Discrete Maximal Parabolic Regularity for Galerkin Finite Element Solutions and their Applications
Maximal parabolic regularity is an important analytical tool and has a number of applications, especially to nonlinear problems and/or optimal control problems when sharp regularity results are required. Recently, there have been a lot of interest in establishing similar results for various time discretization methods. In my talk, I will describe our results for discontinuous Galerkin time schemes and show how such results can be used, for example, in establishing pointwise best approximation estimates for fully discrete Galerkin solutions for parabolic and transient Stokes problems.
New Jersey Institute of Technology
A High Frequency Galerkin Boundary Element Method for Sound-hard Scattering Problems
Considering a plane wave incidence impinging on a two-dimensional convex obstacle, we derive the high-frequency asymptotic expansion of the solution and establish wavenumber explicit estimates on its derivatives. We use these estimates to develop efficient Galerkin approximation spaces for the solution of high-frequency integral equation formulations. Numerical examples validating the theoretical results will be presented.