We propose a Discontinuous Galerkin (DG) scheme for the numerical solution of the Hydrostatic Stokes equations in Oceanography. This new scheme is based on the introduction of the symmetric interior penalty (SIP) technique for the Hydrostatic Stokes mixed variational formulation. Recent research showed that stability of the mixed formulation of Primitive Equations requires LBB (Ladyzhenskaya--Babuška--Brezzi) inf-sup condition and an extra hydrostatic inf-sup restriction relating the pressure and the vertical velocity. This hydrostatic inf-sup condition invalidates usual Stokes continuous finite elements like Taylor-Hood $P_2/P_1$ or bubble $P_{1,b}/P_1$. Here we consider $P_k/P_k$ discontinuous finite elements and, using adequate LBB-like and hydrostatic discrete inf-sup conditions we can demonstrate stability of the SIP DG scheme in the natural energy norm for this problem. Finally, according numerical tests are provided.
Discontinuous Galerkin method for coupling hydrostatic free surface flows to saturated subsurface systems
Friedrich–Alexander University of Erlangen–Nürnberg（埃尔朗根-纽伦堡弗里德里希·亚历山大大学）；Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research（阿尔弗雷德·韦格纳极地与海洋研究所）
This work presents IMplicit-EXplicit (IMEX) formulations for discontinuous Galerkin (DG) discretizations of the compressible Euler equations governing non-hydrostatic atmospheric flows. In particular, we show two different IMEX formulations that not only treat the stiffness due to the governing dynamics but also the domain discretization. We present these formulations for two different equation sets typically employed in atmospheric modeling. For both equation sets, efficient Schur complements are derived and the challenges and remedies for deriving them are discussed. The performance of these…
p-multigrid method for the discontinuous Galerkin discretization of elliptic problems
Hua Loo-Keng Center for Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院华罗庚数学科学研究院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所LSEC）；School of Mathematical Science, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
On Composite Discontinuous Galerkin Method for simulations of electric properties of semiconductor devices
Institute of High Pressure Physics, Polish Academy of Sciences（波兰科学院高压物理研究所）；Department of Mathematics, Computer Science and Mechanics, Warsaw University（华沙大学数学、计算机与力学系）；Interdisciplinary Centre for Materials Modeling, Warsaw University（华沙大学跨学科材料建模中心）
We develop a general convergence theory for adaptive discontinuous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as well as all practically relevant marking strategies. Another key feature of the presented result is, that it holds for penalty parameters only necessary for the standard analysis of the respective scheme. The analysis is based on a quasi interpolation into a newly developed limit space of the adaptively created non-conforming discrete spaces, which enables to generalise the basic convergence result for conforming adaptive finite element method…
Time-space adaptive discontinuous Galerkin method for advection-diffusion equations with non-linear reaction mechanism
Middle East Technical University（中东理工大学）；Institute of Applied Mathematics, Middle East Technical University（中东理工大学应用数学研究所）；Department of Mathematics, Middle East Technical University（中东理工大学数学系）
The stress-strain constitutive law for viscoelastic materials such as soft tissues, metals at high temperature, and polymers, can be written as a Volterra integral equation of the second kind with a \emph{fading memory} kernel. This integral relationship yields current stress for a given strain history and can be used in the momentum balance law to derive a mathematical model for the resulting deformation. We consider such a dynamic linear viscoelastic model problem resulting from using a \textit{Dirichlet-Prony} series of decaying exponentials to provide the fading memory in the Volterra kern…
We study Dual-Primal Isogeometric Tearing and Interconnecting (IETI-DP) solvers for non-conforming multi-patch discretizations of a generalized Poisson problem. We realize the coupling between the patches using a symmetric interior penalty discontinuous Galerkin (SIPG) approach. Previously, we have assumed that the interfaces between patches always consist of whole edges. In this paper, we drop this requirement and allow T-junctions. This extension is vital for the consideration of sliding interfaces, for example between the rotor and the stator of an electrical motor. One critical part for th…
Isogeometric Analysis is a spline-based discretization method to partial differential equations which shows the approximation power of a high-order method. The number of degrees of freedom, however, is as small as the number of degrees of freedom of a low-order method. This does not come for free as the original formulation of Isogeometric Analysis requires a global geometry function. Since this is too restrictive for many kinds of applications, the domain is usually decomposed into patches, where each patch is parameterized with its own geometry function. In simpler cases, the patches can be …
We present a well-balanced nodal discontinuous Galerkin (DG) scheme for compressible Euler equations with gravity. The DG scheme makes use of discontinuous Lagrange basis functions supported at Gauss-Lobatto-Legendre (GLL) nodes together with GLL quadrature using the same nodes. The well-balanced property is achieved by a specific form of source term discretization that depends on the nature of the hydrostatic solution, together with the GLL nodes for quadrature of the source term. The scheme is able to preserve isothermal and polytropic stationary solutions upto machine precision on any mesh …
In this paper, we develop a well-balanced oscillation-free discontinuous Galerkin (OFDG) method for solving the shallow water equations with a non-flat bottom topography. One notable feature of the constructed scheme is the well-balanced property, which preserves exactly the hydrostatic equilibrium solutions up to machine error. Another feature is the non-oscillatory property, which is very important in the numerical simulation when there exist some shock discontinuities. To control the spurious oscillations, we construct an OFDG method with an extra damping term to the existing well-balanced …
Discontinuous Galerkin finite element operator network for solving non-smooth PDEs
Department of Mathematics, Florida State University（数学系，佛罗里达州立大学）；Department of Mathematical Sciences, Seoul National University（数学科学系，首尔国立大学）；Department of AI, Chung-Ang University（人工智能系， Chung-Ang 大学）
Equilibrium or stationary solutions usually proceed through the exact balance between hyperbolic transport terms and source terms. Such equilibrium solutions are affected by truncation errors that prevent any classical numerical scheme from capturing the evolution of small amplitude waves of physical significance. In order to overcome this problem, we compare two commonly adopted strategies: going to very high order and reduce drastically the truncation errors on the equilibrium solution, or design a specific scheme that preserves by construction the equilibrium exactly, the so-called well-bal…
A discontinuous Galerkin method for a new class of Green-Naghdi equations on simplicial unstructured meshes
INSA Toulouse（图卢兹国立应用科学学院）；University of Montpellier, Institut Montpelliérain Alexander Grothendieck and Inria LEMON team（蒙彼利埃大学，亚历山大·格罗滕迪克蒙彼利埃研究所和Inria LEMON团队）
This paper designs and analyzes positivity-preserving well-balanced (WB) central discontinuous Galerkin (CDG) schemes for the Euler equations with gravity. A distinctive feature of these schemes is that they not only are WB for a general known stationary hydrostatic solution, but also can preserve the positivity of the fluid density and pressure. The standard CDG method does not possess this feature, while directly applying some existing WB techniques to the CDG framework may not accommodate the positivity and keep other important properties at the same time. In order to obtain the WB and posi…
The present work develops hybrid multigrid methods for high-order discontinuous Galerkin discretizations of elliptic problems. Fast matrix-free operator evaluation on tensor product elements is used to devise a computationally efficient PDE solver. The multigrid hierarchy exploits all possibilities of geometric, polynomial, and algebraic coarsening, targeting engineering applications on complex geometries. Additionally, a transfer from discontinuous to continuous function spaces is performed within the multigrid hierarchy. This does not only further reduce the problem size of the coarse-grid p…
Numerical approximation of the 3d hydrostatic Navier-Stokes system with free surface
Research School of Earth Sciences, Australian National University（澳大利亚国立大学地球科学研究学院）；Inria Paris（法国国家信息与自动化研究所巴黎分院）；Sorbonne Université, Lab. Jacques-Louis Lions（索邦大学雅克-路易· Lions 实验室）；CEREMA（法国中央环境与市政工程局）；Univ. Paris Diderot, Sorbonne Paris Cité, Institut de Physique du Globe de Paris（巴黎第七大学、索邦巴黎城、巴黎地球物理研究所）
This paper presents a class of novel high-order accurate discontinuous Galerkin (DG) schemes for the compressible Euler equations under gravitational fields. A notable feature of these schemes is that they are well-balanced for a general hydrostatic equilibrium state, and at the same time, provably preserve the positivity of density and pressure. In order to achieve the well-balanced and positivity-preserving properties simultaneously, a novel DG spatial discretization is carefully designed with suitable source term reformulation and a properly modified Harten-Lax-van Leer contact (HLLC) flux.…
