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Multiadaptive Galerkin methods for ODEs III: A priori error estimates

Journal article
Authors Anders Logg
Published in SIAM Journal on Numerical Analysis
Volume 43
Issue 6
Pages 2624-2646
ISSN 0036-1429
Publication year 2006
Published at Department of Mathematical Sciences, Mathematics
Pages 2624-2646
Language en
Links dx.doi.org/10.1137/040604133
Keywords A priori error estimates , Continuous Galerkin , Discontinuous Galerkin , Existence , Individual time steps , Interpolation estimates , Local time steps , mcG(q) , mdG(q) , Multiadaptivity , ODE , Peano kernel theorem , Piecewise smooth , Stability
Subject categories Mathematics, Computational Mathematics

Abstract

The multiadaptive continuous/discontinuous Galerkin methods mcG(q) and mdG(q) for the numerical solution of initial value problems for ordinary differential equations 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. © 2006 Society for Industrial and Applied Mathematics.

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