Multiadaptive Galerkin methods for ODEs. III: A priori error estimates. (English) Zbl 1106.65070
Summary: The multiadaptive continuous/discontinuous Galerkin methods mcG(\(q\)) and mdG(\(q\)) for the numerical solution of initial value problems for ordinary differential equations (ODEs) are based on piecewise polynomial approximation of degree \(q\) on partitions in time with time steps which may vary for different components of the computed solution.
In this paper, we prove general order a priori error estimates for the mcG(\(q\)) and mdG(\(q\)) methods. To prove the error estimates, we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution. The estimates then follow from interpolation estimates, together with stability estimates for the discrete dual solution.
[For part I and II see SIAM J. Sci. Comput. 24, No. 6, 1879–1902 (2003; Zbl 1042.65066) and ibid. 25, No. 4, 1119–1141 (2003; Zbl 1073.65078).]
In this paper, we prove general order a priori error estimates for the mcG(\(q\)) and mdG(\(q\)) methods. To prove the error estimates, we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution. The estimates then follow from interpolation estimates, together with stability estimates for the discrete dual solution.
[For part I and II see SIAM J. Sci. Comput. 24, No. 6, 1879–1902 (2003; Zbl 1042.65066) and ibid. 25, No. 4, 1119–1141 (2003; Zbl 1073.65078).]
MSC:
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
