### MiniSymposium-05: Recent Advances in Finite Element Methods

Abstract: The finite element method is an efficient numerical method to solve partial differential equations, which governs many real-world applications. There have been several new developments in this field recently. This minisymposium intends to create a forum for researches to discuss recent advances in the finite element method.

Organizers:

Saturday, October 19th, 2019 at 16:40 - 18:00 (204 Carver Hall)
16:40 - 17:00 Ahmed Al-Taweel,
University of Arkansas at Little Rock
A P0-P0 Weak Galerkin Finite Element Method for Solving Singularly Perturbed Fraction-Diffusion Problems

This work investigates the lowest-order weak Galerkin nite element (WGFE) method for solving reaction-diffusion equations with singular perturbations in two and three space dimensions. The system of linear equations for the new scheme is positive definite, and one might readily get the well-posedness of the system. Our numerical experiments confirmed our error analysis that our WGFE method of the lowest order could deliver numerical approximations of the order O(h^0.5) and O(h) in H1 and L2 norms, respectively.

17:00 - 17:20 Xiang Wang,
Jilin University
L2 Error Estimate and Superconvergence of High-order Finite Volume Element Methods on Triangular Meshes

We propose a unified framework for the construction of the schemes and analysis of the L2 error estimate for arbitrary k order finite volume element method (FVEM) on triangular meshes. “orthogonal conditions” are proposed to construct the dual meshes, which guarantee the optimal convergence rate with L2 norm of the corresponding FVEM schemes. Moreover, with the orthogonal conditions, we establish the superconvergence theory for quadratic FVEM on triangular meshes.

17:20 - 17:40 Saqib Hussain,
Texas A&M International University
A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

In this article, a weak Galerkin finite element method for Laplace equation using harmonic polynomial space is proposed and analyzed. The idea of using $$P_k$$-harmonic polynomial space instead of the full polynomial space $$P_{k}$$ is to use a much smaller number of basis functions to achieve same accuracy when $$k >= 2$$. The optimal rate of convergence is derived in both $$H^1$$ and $$L^2$$ norms. Numerical experiments have been conducted to verify the theoretical error estimates. In addition, numerical comparisons of using $$P_{2}$$-harmonic polynomial space and using the standard $$P_2$$ polynomial space are presented.

17:40 - 18:00 James Liu,
Colorado State University
An Efficient Numerical Solver for Anisotropic Subdiffusion Problems

In this talk, we present an efficient and robust numerical solver for anisotropic subdiffusion problems, which have not been fully investigated in the literature. The Chebyshev spectral collocation method is utilized for discretization of the spatial Laplacian, whereas a linear interpolation is used for discretizing the fractional order Caputo temporal derivative. This solver is stable and catches the main features of subdiffusion. Numerical experiments are presented to demonstrate the accuracy and efficiency of this new solver.

Sunday, October 20th, 2019 at 10:20 - 11:40 (204 Carver Hall)
10:20 - 10:40 David Lund,
Missouri University of Science and Technology
Fully Kinetic PIFE-PIC Simulations of Plasma Charging at Lunar Craters

This presentation presents a fully-kinetic numerical investigation of plasma charging at lunar carters using the most recently developed Parallel Immersed-Finite-Element Particle-in-Cell (PIFE-PIC) code. This model explicitly includes the lunar regolith layer and the bedrock in the simulation domain, taking into account of the regolith layer thickness and permittivity, and is capable of resolving a non trivial surface terrain or spacecraft configuration. 3-D domain decomposition is used for both particle-push and field-solve steps of the PIC loop to distribute the computation among multiple processors. Simulations are presented to study surface charging and lunar lander charging near the lunar craters.

10:40 - 11:00 Changxin Qiu,
Iowa State University
An Artificial Compressibility Ensemble Method for Stokes-Darcy Model

We propose and analyze an efficient ensemble algorithm with artificial compressibility for fast decoupled computation of multiple realizations of the stochastic Stokes-Darcy model with random hydraulic conductivity (including the one in the interface conditions), source terms, and initial conditions. The solutions are found by solving three smaller decoupled subproblems with two common time-independent coefficient matrices for all realizations, which significantly improves the efficiency for both assembling and solving the matrix systems. The fully coupled Stokes-Darcy system can be first decoupled into two smaller sub-physics problems by the idea of the partitioned time stepping, which reduces the size of the linear systems and allows parallel computing for each sub-physics problem. The artificial compressibility further decouples the velocity and pressure which further reduces storage requirements and improves computational efficiency. We prove the long time stability and the convergence for this new ensemble method.

11:00 - 11:20 Sarah Locke,
Missouri University of Science and Technology
New Proper Orthogonal Decomposition Approximation Theory for PDE Solution Data

When using proper orthogonal decomposition (POD) as a model order reduction technique it is of particular interest to know how these approximations behave. We present refinements and extensions of our earlier results concerning POD projections and approximation errors. We focus on new, exact error formulas for POD approximations with a generalized framework.

11:20 - 11:40 Xiu Ye,
University of Arkansas at Little Rock
Finite Element Methods with Discontinuous Approximations

In this presentation, different finite element methods with discontinuous approximations will be discussed including IPDG, HDG and specially WG finite element methods as well as the relations between them. In addition, new stabilizer free discontinuous finite element methods will be introduced.

Coordinated by
Iowa State University