# Superconvergence of the discontinuous Galerkin method for nonlinear second-order initial-value problems for ordinary differential equations

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4 Citations (Scopus)

### Abstract

In this paper, we propose and analyze a superconvergent discontinuous Galerkin (DG) method for nonlinear second-order initial-value problems for ordinary differential equations. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L2-norm. 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 p-degree DG solutions are O(h2p+1) superconvergent at the downwind points. Finally, we prove that the DG solutions are superconvergent with order p+2 to a particular projection of the exact solutions. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise Pp polynomials with arbitrary p≥1. Computational results indicate that the theoretical orders of convergence and superconvergence are optimal.

Original language English (US) 160-179 20 Applied Numerical Mathematics 115 https://doi.org/10.1016/j.apnum.2017.01.007 Published - May 1 2017

### Fingerprint

Discontinuous Galerkin
Superconvergence
Initial value problems
Discontinuous Galerkin Method
Piecewise Polynomials
Order of Convergence
Galerkin methods
Ordinary differential equations
Initial Value Problem
Ordinary differential equation
A Priori Error Estimates
Optimal Error Estimates
Auxiliary Variables
Arbitrary
Polynomials
Computational Results
Exact Solution
Mesh
Projection
Valid

### Keywords

• A priori error estimates
• Discontinuous Galerkin method
• Nonlinear second-order ordinary differential equations equation
• Superconvergence

### ASJC Scopus subject areas

• Numerical Analysis
• Computational Mathematics
• Applied Mathematics

### Cite this

In: Applied Numerical Mathematics, Vol. 115, 01.05.2017, p. 160-179.

Research output: Contribution to journalArticle

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