Abstract
In this paper we investigate the superconvergence properties of the discontinuous Galerkin method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We show that the discontinuous finite element solution is O(h p+2) superconvergent at the Legendre points on the outflow edge for triangles having one outflow edge. For triangles having two outflow edges the finite element error is O(h p+2) superconvergent at the end points of the inflow edge. Several numerical simulations are performed to validate the theory. In Part II of this work we explicitly write down a basis for the leading term of the error and construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on more general meshes.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 75-113 |
| Number of pages | 39 |
| Journal | Journal of Scientific Computing |
| Volume | 33 |
| Issue number | 1 |
| DOIs | |
| State | Published - Oct 1 2007 |
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Keywords
- Discontinuous Galerkin method
- Hyperbolic problems
- Superconvergence
- Triangular meshes
ASJC Scopus subject areas
- Theoretical Computer Science
- Software
- Engineering(all)
- Computational Theory and Mathematics
Cite this
The discontinuous Galerkin method for two-dimensional hyperbolic problems. Part I : Superconvergence error analysis. / Adjerid, Slimane; Baccouch, Mahboub.
In: Journal of Scientific Computing, Vol. 33, No. 1, 01.10.2007, p. 75-113.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - The discontinuous Galerkin method for two-dimensional hyperbolic problems. Part I
T2 - Superconvergence error analysis
AU - Adjerid, Slimane
AU - Baccouch, Mahboub
PY - 2007/10/1
Y1 - 2007/10/1
N2 - In this paper we investigate the superconvergence properties of the discontinuous Galerkin method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We show that the discontinuous finite element solution is O(h p+2) superconvergent at the Legendre points on the outflow edge for triangles having one outflow edge. For triangles having two outflow edges the finite element error is O(h p+2) superconvergent at the end points of the inflow edge. Several numerical simulations are performed to validate the theory. In Part II of this work we explicitly write down a basis for the leading term of the error and construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on more general meshes.
AB - In this paper we investigate the superconvergence properties of the discontinuous Galerkin method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We show that the discontinuous finite element solution is O(h p+2) superconvergent at the Legendre points on the outflow edge for triangles having one outflow edge. For triangles having two outflow edges the finite element error is O(h p+2) superconvergent at the end points of the inflow edge. Several numerical simulations are performed to validate the theory. In Part II of this work we explicitly write down a basis for the leading term of the error and construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on more general meshes.
KW - Discontinuous Galerkin method
KW - Hyperbolic problems
KW - Superconvergence
KW - Triangular meshes
UR - http://www.scopus.com/inward/record.url?scp=34548401244&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=34548401244&partnerID=8YFLogxK
U2 - 10.1007/s10915-007-9144-x
DO - 10.1007/s10915-007-9144-x
M3 - Article
VL - 33
SP - 75
EP - 113
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
SN - 0885-7474
IS - 1
ER -