L1Solutions to First Order Hyperbolic Equations in Bounded Domains

2003 ◽  
Vol 28 (1-2) ◽  
pp. 381-408 ◽  
Author(s):  
Alessio Porretta ◽  
Julien Vovelle
Keyword(s):  
Hyperbolic Equations ◽  
Bounded Domains ◽  
First Order

Superconvergence of bi-k Degree Time-Space Fully Discontinuous Finite Element for First-Order Hyperbolic Equations

2015 ◽  
Vol 7 (3) ◽  
pp. 323-337 ◽  
Author(s):  
Hongling Hu ◽  
Chuanmiao Chen
Keyword(s):  
Finite Element ◽  
Hyperbolic Equations ◽  
Optimal Convergence Rate ◽  
Galerkin Finite Element ◽  
First Order ◽  
Optimal Convergence ◽  
Time Space ◽  
Correction Technique ◽  
Discontinuous Galerkin Finite Element ◽  
Discontinuous Finite Element

AbstractIn this paper, we present a superconvergence result for the bi-k degree time-space fully discontinuous finite element of first-order hyperbolic problems. Based on the element orthogonality analysis (EOA), we first obtain the optimal convergence order of discontinuous Galerkin finite element solution. Then we use orthogonality correction technique to prove a superconvergence result at right Radau points, which is higher one order than the optimal convergence rate. Finally, numerical results are presented to illustrate the theoretical analysis.

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.

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