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The finite element method for parabolic equations

I. A posteriori error estimation

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Summary

In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.

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Authors and Affiliations

  1. Laboratory of Applied Studies, Division of Computer Research and Technology, National Institutes of Health, 20205, Bethesda, MD, USA

    M. Bieterman

  2. Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    I. Babuška

Authors
  1. M. Bieterman
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  2. I. Babuška
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Additional information

The work was partially supported by ONR Contract N00014-77-C-0623

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Bieterman, M., Babuška, I. The finite element method for parabolic equations. Numer. Math. 40, 339–371 (1982). https://doi.org/10.1007/BF01396451

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  • DOI: https://doi.org/10.1007/BF01396451

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