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An $$hp$$ a priori error analysis of the DG time-stepping for initial value problems. (English) Zbl 1012.65084
This paper examines the error estimates of the discontinuous Galerkin (DG) method for the numerical solution of ordinary differential equations. These methods in fact correspond to the $$L$$-stable Padé methods on the first subdiagonal of the Padé table. The formulation is an $$hp$$ version which is a mix of fixed order, variable time steps ($$p$$ version) and varying order, fixed step ($$h$$ version). Error bounds explicit in the time step and orders are given and it is shown that this approach can give spectral and exponential accuracy for smooth problems and that temporal singularities can be resolved at exponential rates of convergence with geometric time step refinement. The proofs seem to ignore the interpolation errors.

##### MSC:
 65L70 Error bounds for numerical methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
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