MiniSymposium-8: Recent Advances in Numerical Methods for Partial Differential Equations

Numerical methods for partial differential equations play an important role in computational and applied mathematics and its engineering applications. This mini symposium will bring together researchers to discuss the recent advances in this field. The topics of talks include structure-preserved numerical methods, immersed finite element methods for interface problems, high-order numerical methods, discontinuous Galerkin methods, efficient numerical solvers, etc.


Xiaoming He, Missouri University of Sciences and Technology
Xu Zhang, Oklahoma State University


Saturday, October 2, 2021 at 10:20 – 11:40 am (CST) (Session 1)


Amanda Diegel
Mississippi State University
10:20 am

Continuous Data Assimilation and Long-time Accuracy in a C0-IP Method for the Cahn-Hilliard Equation

We propose a numerical approximation method for the Cahn-Hilliard equations that incorporates continuous data assimilation in order to achieve long time accuracy. The method uses a C0 interior penalty spatial discretization of the fourth order Cahn-Hilliard equations, together with a semi-implicit temporal discretization. We prove the method is long time stable and long time accurate, for arbitrarily inaccurate initial conditions, provided enough data measurements are incorporated into the simulation. Numerical experiments illustrate the effectiveness of the method on a benchmark test problem.

Qiao Zhuang
Worcester Polytechnic Institute
10:40 am

Error Analysis of Symmetric Linear/Bilinear Partially Penalized Immersed Finite Element Methods for Helmholtz Interface Problems

This talk presents an error analysis of the symmetric linear/bilinear partially penalized immersed finite element (PPIFE) methods for interface problems of Helmholtz equations. Under the assumption that the exact solution possesses a usual piecewise $H^2$ regularity, the optimal error bounds for the PPIFE solutions are derived in an energy norm and the usual $L^2$ norm. A numerical example is conducted to validate the theoretical conclusions.

Paul Kuberry
Sandia National Lab
11:00 am

Regression Based Approach for Robust Finite Element Analysis on Arbitrary Grids (REBAR)

REBAR is an approach for accurate and robust numerical simulation of partial differential equations for meshes that are of poor quality. Traditional finite element methods use the mesh to both discretize the geometric domain and to define the finite element shape functions. The latter creates a dependence between the quality of the mesh and the properties of the finite element basis that may adversely affect the accuracy of the discretized problem. Our approach breaks this dependence and separates mesh quality from the discretization quality. At the core of the approach is a meshless definition of the shape functions, which limits the purpose of the mesh to representing the geometric domain and integrating the basis functions without having any role in their approximation quality.

A collection of numerical experiments will be presented: strongly coercive elliptic problems, linear elasticity in the compressible regime, and the stationary Stokes problem. Comparison of timestep and accuracy will be made with the continuous Galerkin finite element method.

Yuan Chen
Ohio State University
11:20 am

An Immersed P2-P1 Finite Element Method for Stokes Interface Problems

Stokes interface problems describe multiphase flow with jumps in velocity, pressure, and physical parameters. Simulations of multiphase flow are widely applied in fields of fluid dynamics. Classical finite element method can solve interface problems on meshes designed to be aligned with interfaces. Immersed finite element (IFE) method, on the other hand, allows interfaces to be immersed into elements and works on unfitted meshes such as Cartesian meshes. In this talk, we introduce an immersed finite element method for solving Stokes interface problems. An immersed P2-P1 finite element space is constructed according to the actual interface based on a least square construction. Properties of this space including unisolvence, partition of unity and least square approximation are analyzed. This space is employed in a partially penalized scheme to solve Stokes interface problems. Extra ghost penalty terms are added on both element edges and interface segments for stabilization purpose. Numerical results are provided to demonstrate the behavior of the proposed method. This is a joint work with Dr. Xu Zhang.

Saturday, October 2, 2021 at 4:40 – 6:00 pm (CST) (Session 2)


Cheng Wang
University of Massachusetts-Dartmouth
4:40 pm

A Positivity Preserving, Energy Stable and Convergent Numerical Scheme for the Poisson-Nernst-Planck System

A finite difference numerical scheme is proposed and analyzed for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility, conserved gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the discrete maximum norm bound), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this talk, which demonstrates the robustness of the proposed numerical scheme.

Jea-Hyun Park
University of California-Santa Barbara
5:00 pm

Preconditioned Accelerated Gradient Descent Methods for Locally Lipschitz Smooth Objectives with Applications to the Solution of Nonlinear Partial Differential Equations

We talk about an efficient solver, called preconditioned Nesterov's accelerated gradient descent methods (PAGD) for nonlinear PDEs. We take a continuous model approach involving a second order ordinary differential equation (ODE) as the limiting case of the PAGD to understand how this method efficiently finds the solution to PDEs derived from a certain type of energy. An exponential convergence is discussed under a natural time step restriction for energy stability. A concrete application of the method is also discussed in the context of solving certain nonlinear elliptic PDE using Fourier collocation methods, and several numerical experiments are conducted.

Tulin Kaman
University of Arkansas
5:20 pm

Shock-Turbulence Interactions in Hydrodynamic Instabilities

The shock-turbulence interaction at high Reynolds numbers require high order non-oscillatory numerical schemes to achieve accurate solutions in compressible turbulence. The stable and high-order accurate Weighted Essentially Non-Oscillatory (WENO) scheme shows promising results on the numerical approximations of the compressible turbulent mixing. Turbulent mixing due to the Rayleigh-Taylor instabilities and the effect of numerical schemes on the quantity of interests are investigated. We verify the WENO schemes on two benchmarking tests: the sod shock tube and shock-entropy wave interactions. We also perform numerical studies to investigate the growth rate of the interfaces

Xuejian Li
Missouri University of Science and Technology
5:40 pm

Variational Data Assimilation for Parabolic Interface Equation

In this report we propose and analyze a numerical method of variational data assimilation (VDA) for a second order parabolic interface equation on a two-dimensional bounded domain. By using Tikhonov regularization we formulate the data assimilation problem into an optimization problem. Existence, uniqueness, and stability of the optimal solution are then established. The standard adjoint operation is utilized to derive the first order continuous optimality system. To numerically solve the data assimilation problem, a finite element discretization is designed for spatial approximation and the backward Euler scheme is used in the temporal discretization. The convergence with the optimal error estimate is proved with special attention paid to the recovery of Galerkin orthogonality. In addition, due to the extreme computational cost in VDA, we focus more on reducing the CPU memory requirement and simulation time by developing a variety of efficient algorithms. Finally, numerical results are provided to validate the proposed method.