MS09: Recent development on mathematical and numerical analysis of PDEs and their applications


Session A: Oct.1, 3:10pm-4:30pm, Classroom Building 219
Session B: Oct.1, 5:00pm-6:20pm, Classroom Building 219

3:10pm-3:30pm (Oct 1)

Weizhang Huang,
University of Kansas

A moving mesh finite element method for Bernoulli free boundary problems


A moving mesh finite element method is presented for the numerical solution of Bernoulli free boundary problems. The method first formulates a Bernoulli free boundary problem into a moving boundary problem with explicitly defined velocity for the moving boundary and then solves the latter with a moving mesh PDE method. Examples for both exterior and interior Bernoulli problems will be presented.

3:30pm-3:50pm (Oct 1)
CLB 219

Xinfeng Liu,
University of South Carolina

Fast and efficient numerical methods for a class of PDEs with free boundaries
  For reaction-diffusion equations in irregular domain with moving boundaries, the numerical stability constraints from the reaction and diffusion terms often require very restricted time step size, while complex geometries may lead to difficulties in accuracy when discretizing the high-order derivatives on grid points near the boundary. It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties. Applying an implicit scheme may be able to remove the stability constraints on the time step, however, it usually requires solving a large global system of nonlinear equations for each time step, and the computational cost could be significant. Integration factor (IF) or exponential differencing time (ETD) methods are one of the popular methods for temporal partial differential equations (PDEs) among many other methods. In our paper, we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries. In particular, we rewrite all ETD schemes into a linear combination of specific $\phi$-functions and apply one start-of-the-art algorithm to compute the matrix-vector multiplications, which offers significant computational advantages with adaptive Krylov subspaces. In addition, we extend this method by incorporating the level set method to solve the free boundary problem. The accuracy, stability, and efficiency of the developed method are demonstrated by numerical examples.

3:50pm-4:10pm (Oct 1)
CLB 219

Saqib Hussain,

Texas A&M International University

Analysis of weak Galerkin Finite Element with Supercloseness

In [1], the computational performance of various weak Galerkin finite element in terms of stability, convergence, and supercloseness in explored and numerical results are listed in 31 tables. Some of the phenomena can be explained by the existing theoretical results and the others are to be explained. The main purpose of this paper is to provide a unified theoretical foundation to a class of WG schemes, where elements are used for solving the second order elliptic equations on a triangle grid in 2D. With this unified treatment, all of the existing results become special cases. The theoretical conclusions are corroborated by a number of numerical examples. 

[1]. J. Wang, X. Ye, and S. Zhang, “Numerical investigation on weak Galerkin finite elements,” International Journal of Numerical Analysis and Modeling. 17 (4) (2020), 517 – 531. 

4:10pm-4:30pm (Oct 1)
CLB 219
Jay Mayfield,
University of Arizona

An asymptotic Green's function method for vector wave equations
  We develop an asymptotic Green's function method to numerically solve a vector wave equation. The wavefield is split into its forward and backward propagating parts that can be propagated and presented as an integral with the dyadic Green's function following Huygens' principle. The dyadic Green's function is approximated asymptotically with geometric optics approximations, where its phase and amplitude are determined by an eikonal equation and a recurrent system of transport equations, respectively. The solutions to the eikonal and transport equations are approximated analytically by short-time Taylor series expansions, which leads to a short-time propagator for the wavefield. In order to efficiently evaluate the short-time propagator, its lowrank structure is explored via a randomized QR factorization such that the approximated integral can be computed by fast Fourier transforms. The perfectly matched layer technique is further incorporated to facilitate the computation on a bounded domain of interest. The method has complexity O(tεNlogN) per time step, with N the number of spatial points in the mesh and ε the low rank for a predetermined accuracy tolerance ε>0. Numerical experiments are presented to demonstrate the effectiveness of the method.

5:00pm-5:20pm (Oct 1)
CLB 219
Shuwang Li,
Illinois Institute of Technology

Phase field modeling and computation of vesicle growth and shrinkage
  We study a phase field model for vesicle growth or shrinkage based on osmotic pressure that arises due to a chemical potential gradient. The model consists of an Allen-Cahn-like equation, which describes the phase field evolution, a Cahn-Hilliard-like equation, which simulates the fluid concentration, and a Stokes-like equation, which models the fluid flow. It is mass conserved and surface area constrained during the membrane deformation. Conditions for vesicle growth or shrinkage are analyzed via the common tangent construction. The numerical computing is in two-dimensional space using a nonlinear multigrid method consisting of a FAS method for the PDE system. Convergence test suggests that the global error is of first order in time and of second order in space. Numerical results are demonstrated under no flux boundary conditions and with boundary-driven shear flow respectively.

5:20pm-5:40pm (Oct 1)
CLB 219
Dinh-Liem Nguyen,
Kansas State University

A stable sampling method for imaging of photonic crystals
  We consider the inverse problem of determining the geometry of periodic scattering media from scattered field data. In this talk, we will discuss a sampling type method for solving the inverse problem. This sampling method has a new imaging functional that is simple and easy to implement. The theoretical analysis of the imaging functional is analyzed. Our numerical study shows that this sampling method is more stable than the factorization method and may perform better than the orthogonality sampling method in terms of accuracy. This is joint work with Kale Stahl and Trung Truong.

5:40pm-6:00pm (Oct 1)
CLB 219
Lin Mu,
University of Georgia

Pressure Robust Scheme for Incompressible Flow
  In this talk, we shall introduce the recent development regarding the pressure robust finite element method (FEM) for solving incompressible flow. In this talk, we shall discuss the new divergence preserving schemes in designing the robust numerical schemes for incompressible fluid simulation. Due to the viscosity independence in the velocity approximation, our scheme is robust with small viscosity and/or large permeability, which tackles the crucial computational challenges in fluid simulation. We shall discuss the details in the implementation and theoretical analysis. Several numerical experiments will be tested to validate the theoretical conclusion.

6:00pm-6:20pm (Oct 1)
CLB 219

Xuping Tian,
Iowa State University

Dynamic behavior for a gradient algorithm with energy and momentum

We investigate a novel gradient algorithm, using both energy and momentum (called AGEM), for solving general non-convex optimization problems. The solution properties of the AGEM algorithm, including uniformly boundedness and convergence to critical points, are examined. The dynamic behavior is studied through analysis of a high-resolution ODE system. Such ODE system is nonlinear and obtained by taking the limit of the discrete scheme while keeping the momentum effect through a rescale of the momentum parameter. In particular, we show global well-posedness of the ODE system, time-asymptotic convergence of solution trajectories, and further establish a linear convergence rate for objective functions satisfying the Polyak-Lojasiewicz condition.