Optimal error estimates of the local discontinuous Galerkin method for surface diffusion of graphs on Cartesian meshes

Liangyue Ji; Yan Xu

doi:10.1007/s10915-011-9492-4

Optimal error estimates of the local discontinuous Galerkin method for surface diffusion of graphs on Cartesian meshes

Liangyue Ji, Yan Xu

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

In (Xu and Shu in J. Sci. Comput. 40:375-390, 2009), a local discontinuous Galerkin (LDG) method for the surface diffusion of graphs was developed and a rigorous proof for its energy stability was given. Numerical simulation results showed the optimal order of accuracy. In this subsequent paper, we concentrate on analyzing a priori error estimates of the LDG method for the surface diffusion of graphs. The main achievement is the derivation of the optimal convergence rate k+1 in the L ² norm in one-dimension as well as in multi-dimensions for Cartesian meshes using a completely discontinuous piecewise polynomial space with degree k≥1.

Original language	English (US)
Pages (from-to)	1-27
Number of pages	27
Journal	Journal of Scientific Computing
Volume	51
Issue number	1
DOIs	https://doi.org/10.1007/s10915-011-9492-4
State	Published - Apr 2012
Externally published	Yes

Bibliographical note

Funding Information:
Research of Y. Xu was supported by NSFC grant No. 10971211, No. 11031007, FANEDD No. 200916, FANEDD of CAS, NCET No. 09-0922 and the Fundamental Research Funds for the Central Universities.

Keywords

Error estimates
Local discontinuous Galerkin method
Stability
Surface diffusion of graphs

Publisher link

10.1007/s10915-011-9492-4

Find It

Find It at U of M Libraries

Cite this

@article{a8ac083b6a784e979ea23fdfdd89c9ce,

title = "Optimal error estimates of the local discontinuous Galerkin method for surface diffusion of graphs on Cartesian meshes",

abstract = "In (Xu and Shu in J. Sci. Comput. 40:375-390, 2009), a local discontinuous Galerkin (LDG) method for the surface diffusion of graphs was developed and a rigorous proof for its energy stability was given. Numerical simulation results showed the optimal order of accuracy. In this subsequent paper, we concentrate on analyzing a priori error estimates of the LDG method for the surface diffusion of graphs. The main achievement is the derivation of the optimal convergence rate k+1 in the L 2 norm in one-dimension as well as in multi-dimensions for Cartesian meshes using a completely discontinuous piecewise polynomial space with degree k≥1.",

keywords = "Error estimates, Local discontinuous Galerkin method, Stability, Surface diffusion of graphs",

author = "Liangyue Ji and Yan Xu",

note = "Funding Information: Research of Y. Xu was supported by NSFC grant No. 10971211, No. 11031007, FANEDD No. 200916, FANEDD of CAS, NCET No. 09-0922 and the Fundamental Research Funds for the Central Universities.",

year = "2012",

month = apr,

doi = "10.1007/s10915-011-9492-4",

language = "English (US)",

volume = "51",

pages = "1--27",

journal = "Journal of Scientific Computing",

issn = "0885-7474",

publisher = "Springer New York",

number = "1",

}

TY - JOUR

T1 - Optimal error estimates of the local discontinuous Galerkin method for surface diffusion of graphs on Cartesian meshes

AU - Ji, Liangyue

AU - Xu, Yan

N1 - Funding Information: Research of Y. Xu was supported by NSFC grant No. 10971211, No. 11031007, FANEDD No. 200916, FANEDD of CAS, NCET No. 09-0922 and the Fundamental Research Funds for the Central Universities.

PY - 2012/4

Y1 - 2012/4

N2 - In (Xu and Shu in J. Sci. Comput. 40:375-390, 2009), a local discontinuous Galerkin (LDG) method for the surface diffusion of graphs was developed and a rigorous proof for its energy stability was given. Numerical simulation results showed the optimal order of accuracy. In this subsequent paper, we concentrate on analyzing a priori error estimates of the LDG method for the surface diffusion of graphs. The main achievement is the derivation of the optimal convergence rate k+1 in the L 2 norm in one-dimension as well as in multi-dimensions for Cartesian meshes using a completely discontinuous piecewise polynomial space with degree k≥1.

AB - In (Xu and Shu in J. Sci. Comput. 40:375-390, 2009), a local discontinuous Galerkin (LDG) method for the surface diffusion of graphs was developed and a rigorous proof for its energy stability was given. Numerical simulation results showed the optimal order of accuracy. In this subsequent paper, we concentrate on analyzing a priori error estimates of the LDG method for the surface diffusion of graphs. The main achievement is the derivation of the optimal convergence rate k+1 in the L 2 norm in one-dimension as well as in multi-dimensions for Cartesian meshes using a completely discontinuous piecewise polynomial space with degree k≥1.

KW - Error estimates

KW - Local discontinuous Galerkin method

KW - Stability

KW - Surface diffusion of graphs

UR - http://www.scopus.com/inward/record.url?scp=84857922638&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84857922638&partnerID=8YFLogxK

U2 - 10.1007/s10915-011-9492-4

DO - 10.1007/s10915-011-9492-4

M3 - Article

AN - SCOPUS:84857922638

SN - 0885-7474

VL - 51

SP - 1

EP - 27

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

IS - 1

ER -

Optimal error estimates of the local discontinuous Galerkin method for surface diffusion of graphs on Cartesian meshes

Abstract

Bibliographical note

Keywords

Publisher link

Find It

Other files and links

Fingerprint

Cite this