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A posteriori error bounds and global error control for approximation of ordinary differential equations. (English) Zbl 0820.65052

Using a discontinuous Galerkin finite element method for the integration of initial value problems in ordinary differential equations in conjunction with a discrete Gronwall inequality the author combines a priori and a posteriori analyses of the error to construct a rigorous theory of global error control, using information obtained from computable quantities. The a posteriori analysis he presents suggests a new approach to the theory of adaptive error control useful for stiff problems.
The discussion is concluded by exhibiting the behavior of that error control method in a variety of test problems and by demonstrating the validity of the method in giving realistic estimates of the global error and in detecting superstability in the knee problem.

MSC:

65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A30 Linear ordinary differential equations and systems
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