We present the hp-version of the discontinuous Galerkin method for the numerical solution of delay differential equations with nonlinear vanishing delays and derive error bounds that are explicit in the time steps, the degrees of the approximating polynomials, and the regularity properties of the exact solutions. It is shown that the hp discontinuous Galerkin method exhibits exponential rates of convergence for smooth solutions on uniform meshes, and for nonsmooth solutions on geometrically graded meshes. The theoretical results are illustrated by various numerical examples.
Scopus Subject Areas
- Computational Mathematics
- Applied Mathematics
- Discontinuous Galerkin method
- Nonlinear vanishing delay
- Pantograph delay differential equations
- Spectral and exponential accuracy