### Abstract

In this manuscript we investigate the convergence properties of a minimal dissipation local discontinuous Galerkin(md-LDG) method for two-dimensional diffusion problems on Cartesian meshes. Numerical computations show O(h ^{p+1}) L ^{2} convergence rates for the solution and its gradient and O(h ^{p+2}) superconvergent solutions at Radau points on enriched p-degree polynomial spaces. More precisely, a local error analysis reveals that the leading term of the LDG error for a p-degree discontinuous finite element solution is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, LDG solutions are superconvergent at right Radau points obtained as a tensor product of the shifted roots of the (p+1)-degree right Radau polynomial. For tensor product polynomial spaces, the first component of the solution's gradient is O(h ^{p+2}) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(h ^{p+2}) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Several numerical simulations are performed to validate the theory.

Original language | English (US) |
---|---|

Pages (from-to) | 113-152 |

Number of pages | 40 |

Journal | Journal of Scientific Computing |

Volume | 52 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2012 |

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### Keywords

- Elliptic problems
- Local discontinuous Galerkin method
- Superconvergence

### ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Numerical Analysis
- Engineering(all)
- Computational Theory and Mathematics
- Computational Mathematics
- Applied Mathematics

### Cite this

**A superconvergent local discontinuous Galerkin method for elliptic problems.** / Adjerid, Slimane; Baccouch, Mahboub.

Research output: Contribution to journal › Article

*Journal of Scientific Computing*, vol. 52, no. 1, pp. 113-152. https://doi.org/10.1007/s10915-011-9537-8

}

TY - JOUR

T1 - A superconvergent local discontinuous Galerkin method for elliptic problems

AU - Adjerid, Slimane

AU - Baccouch, Mahboub

PY - 2012/7/1

Y1 - 2012/7/1

N2 - In this manuscript we investigate the convergence properties of a minimal dissipation local discontinuous Galerkin(md-LDG) method for two-dimensional diffusion problems on Cartesian meshes. Numerical computations show O(h p+1) L 2 convergence rates for the solution and its gradient and O(h p+2) superconvergent solutions at Radau points on enriched p-degree polynomial spaces. More precisely, a local error analysis reveals that the leading term of the LDG error for a p-degree discontinuous finite element solution is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, LDG solutions are superconvergent at right Radau points obtained as a tensor product of the shifted roots of the (p+1)-degree right Radau polynomial. For tensor product polynomial spaces, the first component of the solution's gradient is O(h p+2) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(h p+2) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Several numerical simulations are performed to validate the theory.

AB - In this manuscript we investigate the convergence properties of a minimal dissipation local discontinuous Galerkin(md-LDG) method for two-dimensional diffusion problems on Cartesian meshes. Numerical computations show O(h p+1) L 2 convergence rates for the solution and its gradient and O(h p+2) superconvergent solutions at Radau points on enriched p-degree polynomial spaces. More precisely, a local error analysis reveals that the leading term of the LDG error for a p-degree discontinuous finite element solution is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, LDG solutions are superconvergent at right Radau points obtained as a tensor product of the shifted roots of the (p+1)-degree right Radau polynomial. For tensor product polynomial spaces, the first component of the solution's gradient is O(h p+2) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(h p+2) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Several numerical simulations are performed to validate the theory.

KW - Elliptic problems

KW - Local discontinuous Galerkin method

KW - Superconvergence

UR - http://www.scopus.com/inward/record.url?scp=84861459417&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861459417&partnerID=8YFLogxK

U2 - 10.1007/s10915-011-9537-8

DO - 10.1007/s10915-011-9537-8

M3 - Article

AN - SCOPUS:84861459417

VL - 52

SP - 113

EP - 152

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

IS - 1

ER -