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

N1 - Funding Information:
The author would also like to thank the anonymous referees for their constructive comments and remarks which helped improve the quality and readability of the paper. This research was partially supported by the University Committee on Research and Creative Activity (UCRCA Proposal 2018-01-F) at the University of Nebraska at Omaha.
Funding Information:
The author would also like to thank the anonymous referees for their constructive comments and remarks which helped improve the quality and readability of the paper. This research was partially supported by the University Committee on Research and Creative Activity (UCRCA Proposal 2018-01-F ) at the University of Nebraska at Omaha.

PY - 2019/11

Y1 - 2019/11

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

AN - SCOPUS:85065521815

VL - 145

SP - 361

EP - 383

JO - Applied Numerical Mathematics

JF - Applied Numerical Mathematics

SN - 0168-9274

ER -