TY - JOUR
T1 - The discontinuous Galerkin method for general nonlinear third-order ordinary differential equations
AU - Baccouch, Mahboub
N1 - Funding Information:
The authors would like to thank the anonymous reviewers for the valuable comments and suggestions which improve the quality of the paper. This research was supported by the NASA Nebraska Space Grant (Federal Award # 80NSSC20M0112 ) at the University of Nebraska at Omaha.
Publisher Copyright:
© 2021 IMACS
PY - 2021/4
Y1 - 2021/4
N2 - In this paper, we propose an optimally convergent discontinuous Galerkin (DG) method for nonlinear third-order ordinary differential equations. Convergence properties for the solution and for the two auxiliary variables that approximate the first and second derivatives of the solution are established. More specifically, we prove that the method is L2-stable and provides the optimal (p+1)-th order of accuracy for smooth solutions when using piecewise p-th degree polynomials. Moreover, we prove that the derivative of the DG solution is superclose with order p+1 toward the derivative of Gauss-Radau projection of the exact solution. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Several numerical results are provided to confirm the convergence of the proposed scheme.
AB - In this paper, we propose an optimally convergent discontinuous Galerkin (DG) method for nonlinear third-order ordinary differential equations. Convergence properties for the solution and for the two auxiliary variables that approximate the first and second derivatives of the solution are established. More specifically, we prove that the method is L2-stable and provides the optimal (p+1)-th order of accuracy for smooth solutions when using piecewise p-th degree polynomials. Moreover, we prove that the derivative of the DG solution is superclose with order p+1 toward the derivative of Gauss-Radau projection of the exact solution. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Several numerical results are provided to confirm the convergence of the proposed scheme.
KW - A priori error estimates
KW - Discontinuous Galerkin method
KW - Nonlinear third-order ordinary differential equations
KW - Superconvergence
UR - http://www.scopus.com/inward/record.url?scp=85099160062&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85099160062&partnerID=8YFLogxK
U2 - 10.1016/j.apnum.2021.01.003
DO - 10.1016/j.apnum.2021.01.003
M3 - Article
AN - SCOPUS:85099160062
VL - 162
SP - 331
EP - 350
JO - Applied Numerical Mathematics
JF - Applied Numerical Mathematics
SN - 0168-9274
ER -