Numerical Analysis of a Stable Discontinuous Galerkin Scheme for the Hydrostatic Stokes Problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Francisco Guillén-González ◽  
M. Victoria Redondo-Neble ◽  
J. Rafael Rodríguez-Galván
Keyword(s):  
Discontinuous Galerkin ◽  
Vertical Velocity ◽  
Large Scale ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Momentum Equation ◽  
Loss Of Ellipticity ◽  
Numerical Tests ◽  
Discontinuous Galerkin Scheme ◽  
Penalty Technique

AbstractWe propose a Discontinuous Galerkin (DG) scheme for the Hydrostatic Stokes equations. These equations, related to large-scale PDE models in Oceanography, are characterized by the loss of ellipticity of the vertical momentum equation. This fact provides some interesting challenges, such as the design of stable numerical schemes. The new scheme proposed here is based on the SIP penalty technique, with a particular treatment of the vertical velocity. It is well-known that stability of the mixed formulation of Primitive Equations requires, besides the LBB inf-sup condition, an additional hydrostatic inf-sup restriction relating pressure and vertical velocity. This hydrostatic inf-sup condition invalidates stability of usual Stokes stable continuous finite elements like Taylor-Hood 𝒫2/𝒫1 or bubble 𝒫1b/𝒫1. Here we prove stability for our 𝒫k/𝒫k DG scheme. Some novel numerical tests are provided which are in agreement with the previous analysis.

scholarly journals Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity for incompressible flows. Part II

2019 ◽  
Vol 53 (2) ◽  
pp. 503-522 ◽  
Author(s):  
Philip L. Lederer ◽  
Christoph Lehrenfeld ◽  
Joachim Schöberl
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Integration Scheme ◽  
Galerkin Methods ◽  
Robust Analysis ◽  
Reconstruction Operator ◽  
The Impact

The present work is the second part of a pair of papers, considering Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity. The first part mainly dealt with presenting a robust analysis with respect to the mesh size h and the introduction of a reconstruction operator to restore divergence-conformity and pressure robustness (pressure independent velocity error estimates) using a modified force discretization. The aim of this part is the presentation of a high order polynomial robust analysis for the relaxed H(div)-conforming Hybrid Discontinuous Galerkin discretization of the two dimensional Stokes problem. It is based on the recently proven polynomial robust LBB-condition for BDM elements, Lederer and Schöberl (IMA J. Numer. Anal. (2017)) and is derived by a direct approach instead of using a best approximation Céa like result. We further treat the impact of the reconstruction operator on the hp analysis and present a numerical investigation considering polynomial robustness. We conclude the paper presenting an efficient operator splitting time integration scheme for the Navier–Stokes equations which is based on the methods recently presented in Lehrenfeld and Schöberl (Comp. Methods Appl. Mech. Eng. 307 (2016) 339–361) and includes the ideas of the reconstruction operator.

Exponential Time Differencing Gauge Method for Incompressible Viscous Flows

2017 ◽  
Vol 22 (2) ◽  
pp. 517-541 ◽  
Author(s):  
Lili Ju ◽  
Zhu Wang
Keyword(s):  
Stokes Equations ◽  
Stokes Problem ◽  
Three Dimensional ◽  
Viscous Flows ◽  
Momentum Equation ◽  
Compact Representation ◽  
Kinematic Equation ◽  
Exponential Time ◽  
Exponential Time Differencing ◽  
Incompressible Viscous Flows

AbstractIn this paper, we study an exponential time differencing method for solving the gauge system of incompressible viscous flows governed by Stokes or Navier-Stokes equations. The momentum equation is decoupled from the kinematic equation at a discrete level and is then solved by exponential time stepping multistep schemes in our approach. We analyze the stability of the proposed method and rigorously prove that the first order exponential time differencing scheme is unconditionally stable for the Stokes problem. We also present a compact representation of the algorithm for problems on rectangular domains, which makes FFT-based solvers available for the resulting fully discretized system. Various numerical experiments in two and three dimensional spaces are carried out to demonstrate the accuracy and stability of the proposed method.

scholarly journals A simple and accurate discontinuous Galerkin scheme for modeling scalar-wave propagation in media with curved interfaces

Geophysics ◽  
2015 ◽  
Vol 80 (2) ◽  
pp. T83-T89 ◽  
Author(s):  
Xiangxiong Zhang ◽  
Sirui Tan
Keyword(s):  
Discontinuous Galerkin ◽  
Large Scale ◽  
Computational Cost ◽  
Scalar Wave ◽  
Material Interfaces ◽  
Galerkin Scheme ◽  
Discontinuous Galerkin Scheme ◽  
Normal Vectors ◽  
Second Order Convergence ◽  
Conventional Scheme

Conventional high-order discontinuous Galerkin (DG) schemes suffer from interface errors caused by the misalignment between straight-sided elements and curved material interfaces. We have developed a novel DG scheme to reduce those errors. Our new scheme uses the correct normal vectors to the curved interfaces, whereas the conventional scheme uses the normal vectors to the element edge. We modify the numerical fluxes to account for the curved interface. Our numerical modeling examples demonstrate that our new discontinuous Galerkin scheme gives errors with much smaller magnitudes compared with the conventional scheme, although both schemes have second-order convergence. Moreover, our method significantly suppresses the spurious diffractions seen in the results obtained using the conventional scheme. The computational cost of our scheme is similar to that of the conventional scheme. The new DG scheme we developed is, thus, particularly useful for large-scale scalar-wave modeling involving complex subsurface structures.

scholarly journals Efficient Variable-Coefficient Finite-Volume Stokes Solvers

2014 ◽  
Vol 16 (5) ◽  
pp. 1263-1297 ◽  
Author(s):  
Mingchao Cai ◽  
Andy Nonaka ◽  
John B. Bell ◽  
Boyce E. Griffith ◽  
Aleksandar Donev
Keyword(s):  
Finite Volume ◽  
Incompressible Flow ◽  
Variable Coefficient ◽  
Large Scale ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Staggered Grid ◽  
Problem Size ◽  
Pressure System ◽  
Multigrid Algorithm

AbstractWe investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocity pressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 7565-7595], as well; established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

scholarly journals Stable Filtering Procedures for Nodal Discontinuous Galerkin Methods

2021 ◽  
Vol 87 (1) ◽  
Author(s):  
Jan Nordström ◽  
Andrew R. Winters
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Galerkin Methods ◽  
Theoretical Discussion ◽  
Numerical Tests ◽  
Dg Methods ◽  
Theoretical Results ◽  
Filtering Procedure

AbstractWe prove that the most common filtering procedure for nodal discontinuous Galerkin (DG) methods is stable. The proof exploits that the DG approximation is constructed from polynomial basis functions and that integrals are approximated with high-order accurate Legendre–Gauss–Lobatto quadrature. The theoretical discussion re-contextualizes stable filtering results for finite difference methods into the DG setting. Numerical tests verify and validate the underlying theoretical results.

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