Global Superconvergence Estimates of Petrov-Galerkin Methods for Hyperbolic Equations

2010 ◽  
Author(s):  
Bo Yang
Keyword(s):  
Hyperbolic Equations ◽  
Galerkin Methods ◽  
Global Superconvergence

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.

Jacobi Spectral Galerkin and Iterated Methods for Nonlinear Volterra Integral Equation

2016 ◽  
Vol 11 (4) ◽  
Author(s):  
Yin Yang ◽  
Yanping Chen
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Rate Of Convergence ◽  
Volterra Integral Equation ◽  
Galerkin Methods ◽  
Nonlinear Volterra Integral Equation ◽  
L2 Norm ◽  
Spectral Galerkin Method ◽  
Global Superconvergence ◽  
Methods Numerical

In this paper, a Jacobi spectral Galerkin method is developed for nonlinear Volterra integral equations (VIEs) of the second kind. The spectral rate of convergence for the proposed method is established in the L∞-norm and the weighted L2-norm. Global superconvergence properties are discussed by iterated Galerkin methods. Numerical results are presented to demonstrate the effectiveness of the proposed method.

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