### Abstract

In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary-value problems (BVPs) of the form u^{″}=f(x,u,u^{′}), x∈[a,b] subject to some suitable boundary conditions at the endpoints x=a and x=b. We prove optimal L^{2} error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivatives of the LDG solutions are superconvergent with order p+1 toward the derivatives of Gauss-Radau projections of the exact solutions. Moreover, we prove that the LDG solutions are superconvergent with order p+2 toward Gauss-Radau projections of the exact solutions. Finally, we prove, for any polynomial degree p, the (2p+1)th superconvergence rate of the LDG approximations at the upwind or downwind points and for the domain average under quasi-uniform meshes. Our numerical experiments demonstrate optimal rates of convergence and superconvergence. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p≥1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

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

Pages (from-to) | 361-383 |

Number of pages | 23 |

Journal | Applied Numerical Mathematics |

Volume | 145 |

DOIs | |

State | Published - Nov 1 2019 |

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### Keywords

- A priori error estimates
- Local discontinuous Galerkin method
- Nonlinear second-order boundary-value problems
- Superconvergence

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

**Analysis of optimal superconvergence of a local discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems.** / Baccouch, Mahboub.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Analysis of optimal superconvergence of a local discontinuous Galerkin method for nonlinear second-order two-point boundary-value problems

AU - Baccouch, Mahboub

PY - 2019/11/1

Y1 - 2019/11/1

N2 - In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary-value problems (BVPs) of the form u″=f(x,u,u′), x∈[a,b] subject to some suitable boundary conditions at the endpoints x=a and x=b. We prove optimal L2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivatives of the LDG solutions are superconvergent with order p+1 toward the derivatives of Gauss-Radau projections of the exact solutions. Moreover, we prove that the LDG solutions are superconvergent with order p+2 toward Gauss-Radau projections of the exact solutions. Finally, we prove, for any polynomial degree p, the (2p+1)th superconvergence rate of the LDG approximations at the upwind or downwind points and for the domain average under quasi-uniform meshes. Our numerical experiments demonstrate optimal rates of convergence and superconvergence. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p≥1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

AB - In this paper, we investigate the convergence and superconvergence properties of a local discontinuous Galerkin (LDG) method for nonlinear second-order two-point boundary-value problems (BVPs) of the form u″=f(x,u,u′), x∈[a,b] subject to some suitable boundary conditions at the endpoints x=a and x=b. We prove optimal L2 error estimates for the solution and for the auxiliary variable that approximates the first-order derivative. The order of convergence is proved to be p+1, when piecewise polynomials of degree at most p are used. We further prove that the derivatives of the LDG solutions are superconvergent with order p+1 toward the derivatives of Gauss-Radau projections of the exact solutions. Moreover, we prove that the LDG solutions are superconvergent with order p+2 toward Gauss-Radau projections of the exact solutions. Finally, we prove, for any polynomial degree p, the (2p+1)th superconvergence rate of the LDG approximations at the upwind or downwind points and for the domain average under quasi-uniform meshes. Our numerical experiments demonstrate optimal rates of convergence and superconvergence. Our proofs are valid for arbitrary regular meshes using piecewise polynomials of degree p≥1 and for the classical sets of boundary conditions. Several computational examples are provided to validate the theoretical results.

KW - A priori error estimates

KW - Local discontinuous Galerkin method

KW - Nonlinear second-order boundary-value problems

KW - Superconvergence

UR - http://www.scopus.com/inward/record.url?scp=85065521815&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065521815&partnerID=8YFLogxK

U2 - 10.1016/j.apnum.2019.05.003

DO - 10.1016/j.apnum.2019.05.003

M3 - Article

VL - 145

SP - 361

EP - 383

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -