### Abstract

In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L^{2} error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p+1)-th order convergent in the L^{2}-norm, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solution and its derivatives up to order three are O(h^{2p−2}) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for P^{p} polynomials with degree p≥3, and for the classical boundary conditions.

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

Pages (from-to) | 91-115 |

Number of pages | 25 |

Journal | Applied Numerical Mathematics |

Volume | 137 |

DOIs | |

State | Published - Mar 2019 |

### Fingerprint

### Keywords

- A priori error estimates
- Fourth-order boundary-value problems
- Superconvergence
- Ultra weak discontinuous Galerkin method
- Upwind and downwind points

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

*Applied Numerical Mathematics*,

*137*, 91-115. https://doi.org/10.1016/j.apnum.2018.11.011

**Optimal error estimates and superconvergence of an ultra weak discontinuous Galerkin method for fourth-order boundary-value problems.** / Baccouch, Mahboub; Temimi, Helmi; Ben-Romdhane, Mohamed.

Research output: Contribution to journal › Article

*Applied Numerical Mathematics*, vol. 137, pp. 91-115. https://doi.org/10.1016/j.apnum.2018.11.011

}

TY - JOUR

T1 - Optimal error estimates and superconvergence of an ultra weak discontinuous Galerkin method for fourth-order boundary-value problems

AU - Baccouch, Mahboub

AU - Temimi, Helmi

AU - Ben-Romdhane, Mohamed

PY - 2019/3

Y1 - 2019/3

N2 - In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L2 error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p+1)-th order convergent in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solution and its derivatives up to order three are O(h2p−2) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with degree p≥3, and for the classical boundary conditions.

AB - In this paper, we study the convergence and superconvergence properties of an ultra weak discontinuous Galerkin (DG) method for linear fourth-order boundary-value problems (BVPs). We prove several optimal L2 error estimates for the solution and its derivatives up to third order. In particular, we prove that the DG solution is (p+1)-th order convergent in the L2-norm, when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solution and its derivatives up to order three are O(h2p−2) superconvergent at either the downwind points or upwind points. Numerical examples demonstrate that the theoretical rates are sharp. We also observed optimal rates of convergence and superconvergence even for nonlinear BVPs. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with degree p≥3, and for the classical boundary conditions.

KW - A priori error estimates

KW - Fourth-order boundary-value problems

KW - Superconvergence

KW - Ultra weak discontinuous Galerkin method

KW - Upwind and downwind points

UR - http://www.scopus.com/inward/record.url?scp=85057542659&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85057542659&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2018.11.011

DO - 10.1016/j.apnum.2018.11.011

M3 - Article

AN - SCOPUS:85057542659

VL - 137

SP - 91

EP - 115

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -