scholarly journals On the Suboptimality of the P-Version Discontinuous Galerkin Methods for First Order Hyperbolic Problems

2021 ◽  
Author(s):  
Z. Dong ◽  
L. Mascotto
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Galerkin Methods ◽  
Hyperbolic Problems ◽  
First Order

scholarly journals DISCONTINUOUS GALERKIN METHODS FOR FIRST-ORDER HYPERBOLIC PROBLEMS

2004 ◽  
Vol 14 (12) ◽  
pp. 1893-1903 ◽  
Author(s):  
F. BREZZI ◽  
L. D. MARINI ◽  
E. SÜLI
Keyword(s):  
Discontinuous Galerkin Methods ◽  
Hyperbolic Equations ◽  
Galerkin Methods ◽  
Penalty Parameter ◽  
Hyperbolic Problems ◽  
First Order ◽  
Combined Use ◽  
Stabilization Procedure ◽  
Diffusive Term ◽  
Dg Methods

The main aim of this paper is to highlight that, when dealing with DG methods for linear hyperbolic equations or advection-dominated equations, it is much more convenient to write the upwind numerical flux as the sum of the usual (symmetric) average and a jump penalty. The equivalence of the two ways of writing is certainly well known (see e.g. Ref. 4); yet, it is very widespread not to consider upwinding, for DG methods, as a stabilization procedure, and too often in the literature the upwind form is preferred in proofs. Here, we wish to underline the fact that the combined use of the formalism of Ref. 3 and the jump formulation of upwind terms has several advantages. One of them is, in general, to provide a simpler and more elegant way of proving stability. The second advantage is that the calibration of the penalty parameter to be used in the jump term is left to the user (who can think of taking advantage of this added freedom), and the third is that, if a diffusive term is present, the two jump stabilizations (for the generalized upwinding and for the DG treatment of the diffusive term) are often of identical or very similar form, and this can also be turned to the user's advantage.

scholarly journals Extendible and Efficient Python Framework for Solving Evolution Equations with Stabilized Discontinuous Galerkin Methods

2021 ◽  
Author(s):  
Andreas Dedner ◽  
Robert Klöfkorn
Keyword(s):  
Partial Differential Equations ◽  
Discontinuous Galerkin ◽  
User Interfaces ◽  
Discontinuous Galerkin Methods ◽  
Evolution Equations ◽  
Nonlinear Partial Differential Equations ◽  
Galerkin Methods ◽  
First Order ◽  
Wide Range ◽  
Dg Method

AbstractThis paper discusses a Python interface for the recently published Dune-Fem-DG module which provides highly efficient implementations of the discontinuous Galerkin (DG) method for solving a wide range of nonlinear partial differential equations (PDEs). Although the C++ interfaces of Dune-Fem-DG are highly flexible and customizable, a solid knowledge of C++ is necessary to make use of this powerful tool. With this work, easier user interfaces based on Python and the unified form language are provided to open Dune-Fem-DG for a broader audience. The Python interfaces are demonstrated for both parabolic and first-order hyperbolic PDEs.

Penalty-free discontinuous Galerkin methods for incompressible Navier–Stokes equations

2014 ◽  
Vol 24 (06) ◽  
pp. 1217-1236 ◽  
Author(s):  
Beatrice Riviere ◽  
Shirin Sardar
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Stokes Equations ◽  
A Priori ◽  
Navier Stokes ◽  
Galerkin Methods ◽  
Navier Stokes Equations ◽  
First Order ◽  
Priori Error Estimates ◽  
The Stability

A first-order discontinuous Galerkin method is proposed for solving the steady-state incompressible Navier–Stokes equations. The stability of this penalty-free method is obtained by locally enriching the discrete space with a quadratic polynomial. A priori error estimates are derived. Numerical examples confirm the theoretical convergence.

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