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A Posteriori Error Analysis and Adaptive Methods for Parabolic Problems

In: Recent Progress in Computational and Applied PDES

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  • Zhiming Chen

    (Chinese Academy of Sciences, LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences)

Abstract

We report the recent progress in deriving sharp a posteriori error estimates for linear and nonlinear parabolic problems. We show how to use special properties of a linear dual problem in non-divergence form with vanishing diffusion and strong advection to derive L 1 L 1 norm estimate for the continuous casting problem. This estimate exhibits a mild explicit dependence on velocity. We next use a direct energy estimate method to develop an efficient and reliable a posteriori error estimator for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. A convergent adaptive algorithm with variable time-step sizes and space meshes is proposed and studied which, at each time step, delays the mesh coarsening until the final iteration of the adaptive procedure, allowing only mesh and time-step size refinements before. The key ingredient in the convergence analysis is a new coarsening strategy.

Suggested Citation

  • Zhiming Chen, 2002. "A Posteriori Error Analysis and Adaptive Methods for Parabolic Problems," Springer Books, in: Tony F. Chan & Yunqing Huang & Tao Tang & Jinchao Xu & Long-An Ying (ed.), Recent Progress in Computational and Applied PDES, pages 145-156, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4615-0113-8_10
    DOI: 10.1007/978-1-4615-0113-8_10
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