Abstract : This minisymposium is on the non-standard finite element methods, including mixed finite element methods, non-conforming finite elements methods, discontinuous Galerkin methods, virtual element methods, weak Galerkin methods, and so on. This minisymposium intended to focus on the latest research progress in the field of numerical methods of partial differential equations. It is expected that through the frontier progress report, scholars engaged in the research of numerical methods of partial differential equations can be exposed to the core problems in the field of mathematical theory, numerical methods, and practical applications.
Organizer(s) : Carsten Carstensen, Jun Hu, Ran Zhang
[04214] A posteriori error estimation for a C1-virtual element method of Kirchhoff plates
Author(s) :
Jianguo Huang (Shanghai Jiao Tong University)
Abstract : A residual-type a posteriori error estimation is developed for a $C^1$-conforming virtual element method (VEM) to solve a Kirchhoff plate bending problem. As an outcome of the error estimator, an adaptive VEM is introduced using the mesh refinement strategy with the one-hanging-node rule. A series of numerical results are performed to verify the efficiency of the method. This is a joint work with Mingqing Chen and Sen Lin from Shanghai Jiao Tong University.
[03654] Stabilization-Free Virtual Element Methods
Author(s) :
Xuehai Huang (Shanghai University of Finance and Economics)
Abstract : Stabilization-free virtual element methods (VEMs) in arbitrary degree of polynomial are developed for second order elliptic problems, including a nonconforming VEM in arbitrary dimension and a conforming VEM in two dimensions. The key is to construct local $H(\div)$-conforming macro finite element spaces such that the associated $L^2$ projection of the gradient of virtual element functions is computable, and the $L^2$ projector has a uniform lower bound on the gradient of virtual element function spaces in $L^2$ norm. Optimal error estimates are derived for these stabilization-free VEMs. Numerical experiments are provided to test the stabilization-free VEMs.
[03806] Discontinuous Galerkin methods for magnetic advection-diffusion problems
Author(s) :
Jindong Wang (Peking University)
Shuonan Wu (Peking University)
Abstract : We devise and analyze a class of the primal discontinuous Galerkin methods for magnetic advection-diffusion problems based on the weighted-residual approach. In addition to the upwind stabilization, we find a new mechanism under the vector case that provides more flexibility in constructing the schemes. For the more general Friedrichs system, we show the stability and optimal error estimate, which boil down to two core ingredients -- the weight function and the special projection -- that contain information of advection. Numerical experiments are provided to verify the theoretical results.
[03492] Some finite element divdiv complexes in three dimensions
Format : Talk at Waseda University
Author(s) :
Rui Ma (Beijing Institute of Technology)
Abstract : This talk will present two families of finite element divdiv complexes on tetrahedral grids and one family on cuboid grids. They can be used to discretize the linearized Einstein-Bianchi system.
[04728] Adaptive FEM for Helmholtz equation with large wave number
Author(s) :
Haijun Wu (Nanjing University)
Songyao Duan (Nanjing University)
Abstract : A posteriori upper and lower bounds are derived for the finite element method (FEM) for the Helmholtz equation with large wavenumber. It is proved rigorously that the standard residual type error estimator seriously underestimates the true error of the FE solution for the mesh size $h$ in the preasymptotic regime, which is first observed by [Babuska,~et~al., A posteriori error estimation for finite element solutions of Helmholtz equation. Part I, Int. J. Numer. Meth. Engrg. 40, 3443--3462 (1997)] for a one dimensional problem. By establishing an equivalence relationship between the error estimators for the FE solution and the corresponding elliptic projection of the exact solution, an adaptive algorithm is proposed and its convergence and quasi-optimality are proved under the condition that $k^{2p+1}h_0^{2p}$ is sufficiently small, where $k$ is the wavenumber, $h_0$ is the initial mesh size.
[03581] Stable Finite Element Scheme for Dynamic Ginzburg-Landau Equations
Author(s) :
Limin Ma (Wuhan University)
Abstract : We propose a decoupled numerical scheme of the time-dependent Ginzburg--Landau equations under the temporal gauge. The maximum bound principle of the order parameter and the energy dissipation law in the discrete sense are proved, which can guarantee the stability and validity of the numerical simulations, and further facilitate the adoption of adaptive time-stepping strategy. An optimal error estimate of the proposed scheme is also proved and verified by numerical examples.
[03508] Local bounded commuting projection operators for discrete finite element complexes
Author(s) :
Ting Lin (Peking University)
Abstract : Local bounded commuting projection operators are an important tool in the analysis of finite element exterior and mixed finite element methods. However, so far only those of the standard finite element spaces have been discussed. In this talk, I will introduce the construction of local bounded commuting projection operators of the discrete finite element complexes, with some possible applications. The techniques developed here also give us a new perspective on the construction of finite element complexes.
[04168] The weak Galerkin method for elliptic eigenvalue problems
Author(s) :
Qilong Zhai (Jilin University)
Abstract : In this report, we propose and analyze the elliptic eigenvalue problems by using the weak Galerkin method. In contrast to the conforming finite element method, the lower bounds of eigenvalues are considered. We prove that the weak Galerkin method produces asymptotic lower bounds by using the high order polynomials, and produces guaranteed lower bounds by using the lowest order polynomials. Some numerical acceleration techniques are also applied to the weak Galerkin method, and the numerical experiments are presented to verify the theoretical analysis.
