Abstract
In this paper, we propose and analyze an efficient and reliable a posteriori error estimator of residual-type for the discontinuous Galerkin (DG) method applied to nonlinear second-order initial-value problems for ordinary differential equations. This estimator is simple, efficient, and asymptotically exact. We use our recent optimal L2 error estimates and superconvergence results of Baccouch [15] to show that the significant parts of the DG discretization errors are proportional to the (p+1)-degree right Radau polynomial, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimator which is obtained by solving a local residual problem with no initial condition on each element. We prove that, for smooth solutions, the proposed a posteriori error estimator converges to the actual error in the L2-norm with order of convergence p+2. Computational results indicate that the theoretical order of convergence is sharp. By adding the a posteriori error estimate to the DG solution, we obtain a post-processed approximation which superconverges with order p+2 in the L2-norm. Moreover, we demonstrate the effectiveness of the this error estimator. Finally, we present a local adaptive mesh refinement (AMR) procedure that makes use of our local a posteriori error estimate. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p≥1. Several numerical results are presented to validate the theoretical results.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 18-37 |
| Number of pages | 20 |
| Journal | Applied Numerical Mathematics |
| Volume | 121 |
| DOIs | |
| State | Published - Nov 2017 |
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Keywords
- A posteriori error estimates
- Adaptive mesh refinement
- Discontinuous Galerkin method
- Nonlinear second-order ordinary differential equations
- Superconvergence
ASJC Scopus subject areas
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics
Cite this
A posteriori error estimates and adaptivity for the discontinuous Galerkin solutions of nonlinear second-order initial-value problems. / Baccouch, Mahboub.
In: Applied Numerical Mathematics, Vol. 121, 11.2017, p. 18-37.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - A posteriori error estimates and adaptivity for the discontinuous Galerkin solutions of nonlinear second-order initial-value problems
AU - Baccouch, Mahboub
PY - 2017/11
Y1 - 2017/11
N2 - In this paper, we propose and analyze an efficient and reliable a posteriori error estimator of residual-type for the discontinuous Galerkin (DG) method applied to nonlinear second-order initial-value problems for ordinary differential equations. This estimator is simple, efficient, and asymptotically exact. We use our recent optimal L2 error estimates and superconvergence results of Baccouch [15] to show that the significant parts of the DG discretization errors are proportional to the (p+1)-degree right Radau polynomial, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimator which is obtained by solving a local residual problem with no initial condition on each element. We prove that, for smooth solutions, the proposed a posteriori error estimator converges to the actual error in the L2-norm with order of convergence p+2. Computational results indicate that the theoretical order of convergence is sharp. By adding the a posteriori error estimate to the DG solution, we obtain a post-processed approximation which superconverges with order p+2 in the L2-norm. Moreover, we demonstrate the effectiveness of the this error estimator. Finally, we present a local adaptive mesh refinement (AMR) procedure that makes use of our local a posteriori error estimate. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p≥1. Several numerical results are presented to validate the theoretical results.
AB - In this paper, we propose and analyze an efficient and reliable a posteriori error estimator of residual-type for the discontinuous Galerkin (DG) method applied to nonlinear second-order initial-value problems for ordinary differential equations. This estimator is simple, efficient, and asymptotically exact. We use our recent optimal L2 error estimates and superconvergence results of Baccouch [15] to show that the significant parts of the DG discretization errors are proportional to the (p+1)-degree right Radau polynomial, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimator which is obtained by solving a local residual problem with no initial condition on each element. We prove that, for smooth solutions, the proposed a posteriori error estimator converges to the actual error in the L2-norm with order of convergence p+2. Computational results indicate that the theoretical order of convergence is sharp. By adding the a posteriori error estimate to the DG solution, we obtain a post-processed approximation which superconverges with order p+2 in the L2-norm. Moreover, we demonstrate the effectiveness of the this error estimator. Finally, we present a local adaptive mesh refinement (AMR) procedure that makes use of our local a posteriori error estimate. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p≥1. Several numerical results are presented to validate the theoretical results.
KW - A posteriori error estimates
KW - Adaptive mesh refinement
KW - Discontinuous Galerkin method
KW - Nonlinear second-order ordinary differential equations
KW - Superconvergence
UR - http://www.scopus.com/inward/record.url?scp=85021635359&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85021635359&partnerID=8YFLogxK
U2 - 10.1016/j.apnum.2017.06.001
DO - 10.1016/j.apnum.2017.06.001
M3 - Article
AN - SCOPUS:85021635359
VL - 121
SP - 18
EP - 37
JO - Applied Numerical Mathematics
JF - Applied Numerical Mathematics
SN - 0168-9274
ER -