Abstract : This minisymposium concentrates on the recent developments in numerical approximations to partial differential equations by hybridizable discontinuous Galerkin methods and other related schemes such as mixed methods, discontinuous Galerkin methods, and hybrid high-order methods. The scope of the talks is open to mathematical theory and applications of the methods in science and computational engineering, including the development of new hybridizable discontinuous Galerkin methods, Hamiltonian structure-preserving methods, grid adaptivity schemes, linear and nonlinear iterative methods, efficient implementations on emerging computer architectures, and robust shock capturing methods.
Organizer(s) : Jay Gopalakrishnan, Cuong-Ngoc Nguyen, Jaime Peraire, Manuel A. Sanchez
[03873] Output-Adaptive Hybridized Discontinuous Finite Elements for Efficient Flow Computations
Format : Talk at Waseda University
Author(s) :
Krzysztof Fidkowski (University of Michigan)
Abstract : Discontinuous Galerkin (DG) methods have enabled accurate computations of complex flowfields, yet their memory footprint and computational costs are large. Hybridized discontinuous Galerkin (HDG/EDG) methods reduce the number of globally-coupled degrees of freedom by decoupling elements and stitching them together through weak flux continuity. However, these methods have not risen to nearly the same level of popularity as DG, and in this work we outline reasons why and demonstrate benefits in an output-based adaptive setting.
[04227] Towards boundary conditions for HDG methods for direct aeroacoustic computations
Format : Talk at Waseda University
Author(s) :
Philip Lukas Lederer (University of Twente)
Jan Ellmenreich (TU Wien)
Abstract : We focus on direct aeroacoustic simulations using hybrid discontinuous Galerkin methods (HDG) for the approximation of the compressible Navier-Stokes equations. A crucial factor for accurate results is the proper handling of artificial boundary conditions, focusing on the reflectivity of the acoustic waves. We discuss various approaches such as LODI relations and extensions of characteristic Navier-Stokes boundary conditions (NSCBC) for HDG methods. Numerical results are presented and discussed in detail.
[04312] Hybrid discontinuous Galerkin methods on multiple levels
Format : Talk at Waseda University
Author(s) :
Guido Kanschat (Heidelberg University)
Peipei Lu (Soochow University)
Roland Maier (University of Jena)
Andreas Rupp (LUT University)
Abstract : Hybrid finite element schemes approximate the trace of an unknown solution to a partial differential equation on the mesh skeleton (the union of the faces in a mesh). Afterward, these schemes use this trace approximation to recover the primal and dual unknowns. This strategy has several advantages over standard finite elements, such as an enhanced order of convergence, smaller symmetric positive systems of linear equations, and preservation of physically meaningful quantities.
Virtually all of the available multilevel (multigrid and multiscale) techniques for finite elements exploit the fact that their test and trial spaces are nested. However, the test and trial spaces are not nested for hybrid finite elements because their main approximate lives on the mesh's skeleton, which grows with refinement. Thus, the major obstacle that blocks the way to practical multigrid methods comprises the construction of stable mesh transfer operators. Several recent works have addressed this issue by exchanging the hybrid formulation for a non-hybrid formulation in a preliminary step. Afterward, these approaches use the available multilevel strategies for non-hybrid finite elements.
We proceed differently and preserve the advantageous properties of hybrid finite elements on all mesh levels. To this end, we devise stable mesh transfer operators and provide relevant convergence results for our hybrid, multilevel finite elements. This approach allows us to use identical discretizations on all mesh levels (homogeneous strategy) instead of the heterogeneous methods that use different discretizations on different mesh levels.
[04720] Multigrid for HDG
Format : Talk at Waseda University
Author(s) :
Guosheng Fu (University of Notre Dame)
Wenzheng Kuang (University of Notre Dame)
Abstract : We present optimal geometric and algebraic multigrid preconditioners for low-order and high-order HDG schemes for diffusion and Stokes problems. The algebraic multigrid is based on Jinchao Xu's auxiliary space preconditioning framework, while the geometric multigrid is based on the close connection between the lowest-order HDG scheme with the nonconforming Cruzeix-Raviart method. This is a joint work with Wenzheng Kuang from Notre Dame.
00919 (2/3) : 4C @E710 [Chair: Jay Gopalakrishnan]
[04456] Combining finite element space-discretizations with symplectic time-marching schemes for linear Hamiltonian systems
Format : Talk at Waseda University
Author(s) :
Manuel Sanchez (Pontificia Universidad Catolica de Chile)
Bernardo Cockburn (University of Minnesota)
Shukai Du (University of Wisconsin-Madision)
Abstract : We provide a short introduction to the devising of a special type of methods for numerically approximating the solution of Hamiltonian partial differential equations. These methods use Galerkin space-discretizations which result in a system of ODEs displaying a discrete version of the Hamiltonian structure of the original system. The resulting system of ODEs is then discretized by a symplectic time-marching method. This combination results in high-order accurate, fully discrete methods which can preserve the invariants of the Hamiltonian defining the ODE system. We restrict our attention to linear Hamiltonian systems, as the main results can be obtained easily and directly, and are applicable to many Hamiltonian systems of practical interest including acoustics, elastodynamics, and electromagnetism. After a brief description of the Hamiltonian systems of our interest, we provide a brief introduction to symplectic time-marching methods for linear systems of ODEs which does not require any background on the subject. We consider the case of finite-element space discretizations. The emphasis is placed on the conservation properties of the fully discrete schemes.
[04725] HDG method for elliptic interface problems and industrial application
Format : Talk at Waseda University
Author(s) :
Masaru Miyashita
Norikazu Saito (The University of Tokyo)
Abstract : We propose a hybridized discontinuous Galerkin (HDG) method to solve the interface problem for elliptic equations. We succeed in deriving optimal order error estimates in both the HDG norm and the L2 norm under low regularity assumptions for solutions such as $u|_{Ω_1} ∈ H^{1+s}(Ω_1)$ and $u|_{Ω_2} ∈ H^{1+s}(Ω_2)$ for some $s ∈ (1/2, 1]$. Numerical examples support our theoretical results. Then, we show an example of developing plasma equipment using the proposed method.
[04892] A C0 interior penalty method for mth-Laplace equation
Format : Online Talk on Zoom
Author(s) :
Weifeng Qiu (City University of Hong Kong)
Huangxin Chen (Xiamen University)
Jingzhi Li (SUSTech)
Abstract : I will talk about a C0 interior penalty method for the mth-Laplace equation.
[05082] Discontinuous Galerkin Methods for High Speed Flows
Format : Online Talk on Zoom
Author(s) :
Ngoc Cuong Nguyen (Massachusetts Institute of Technology)
Jaime Peraire (Massachusetts Institute of Technology)
Jordi Perez (Massachusetts Institute of Technology)
Loek Van Heyningen (Massachusetts Institute of Technology)
Abstract : We present discontinuous Galerkin (DG) methods for high-speed flows with particular focus on transition, turbulence, and shock capturing. We describe LDG, HDG, EDG methods and parallel iterative solvers with matrix-free preconditioners. We develop an adaptive viscosity regularization method for capturing shocks by minimizing the artifial viscosity field while enforcing smoothness contraints on the numerical solution. We present numerical results to demonstrate the DG methods and our shock capturing scheme on transonic, supersonic, and hypersonic flows.
[03033] A C0 interior penalty method for mth-Laplace equation
Format : Online Talk on Zoom
Author(s) :
Weifeng Qiu (City University of Hong Kong)
Huangxin Chen (Xiamen University)
Jingzhi Li (Southern University of Science and Technology)
Abstract : In this paper, we propose a 𝐶0 interior penalty method for 𝑚th-Laplace equation on bounded Lipschitz polyhedral domain in R𝑑, where 𝑚 and 𝑑 can be any positive integers. The standard 𝐻1-conforming piecewise 𝑟-th order polynomial space is used to approximate the exact solution 𝑢, where 𝑟 can be any integer greater than or equal to 𝑚. Unlike the interior penalty method in Gudi and Neilan [IMA J. Numer. Anal. 31 (2011) 1734–1753], we avoid computing 𝐷^𝑚 of numerical solution on each element and high order normal derivatives of numerical solution along mesh interfaces. Therefore our method can be easily implemented. After proving discrete 𝐻𝑚-norm bounded by the natural energy semi-norm associated with our method, we manage to obtain stability and optimal convergence with respect to discrete 𝐻𝑚-norm.
