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Second-order finite element approximations and a posteriori error estimation for two-dimensional parabolic systems

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Summary

We discuss an adaptive local refinement finite element method of lines for solving vector systems of parabolic partial differential equations on two-dimensional rectangular regions. The partial differential system is discretized in space using a Galerkin approach with piecewise eight-node serendipity functions. An a posteriori estimate of the spatial discretization error of the finite element solution is obtained using piecewise fifth degree polynomials that vanish on the edges of the rectangular elements of a grid. Ordinary differential equations for the finite element solution and error estimate are integrated in time using software for stiff differential systems. The error estimate is used to control a local spatial mesh refinement procedure in an attempt to keep a global measure of the error within prescribed limits. Examples appraising the accuracy of the solution and error estimate and the computational efficiency of the procedure relative to one using bilinear finite elements are presented.

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References

  1. Adjerid, S., Flaherty, J.E.: A Moving Finite Element Method with Error Estimation and Refinement for One-Dimensional Time Dependent Partial Differential Equations. SIAM J. Numer. Anal.23, 778–796 (1986)

    Google Scholar 

  2. Adjerid, S., Flaherty, J.E.: A Moving Mesh Finite Element Method with Local Refinement for Parabolic Partial Differential Equations. Comput. Methods Appl. Mech. Eng.56, 3–26 (1986)

    Google Scholar 

  3. Adjerid, S., Flaherty, J.E.: A Local Refinement Finite Element Method for Two-Dimensional Parabolic Systems. Tech. Rep. No. 86-7, Department of Computer Science, Rensselaer Polytechnic Institute 1986

  4. Adjerid, S., Flaherty, J.E.: Local Refinement Finite Element Method on Stationary and Moving Meshes for One-Dimensional Parabolic Systems. 1988 (to appear)

  5. Adjerid, S., Flaherty, J.E.: Adaptive Finite Element Methods for Parabolic Systems in One and Two Space Dimensions. Trans. Fourth Army Conf. Appl. Math. Comput. ARO Report 87-1. pp. 1077–1098 U.S. Army Research Office 1987

  6. Bank, R.E.: PLTMG User's Guide, June 1981 Version. Technical Report, Department of Mathematics, University of California at San Diego 1982

  7. Bank, R.E., Sherman, A.: An Adaptive Multi-Level Method for Elliptic Boundary Value Problems. Computing26, 91–105 (1981)

    Google Scholar 

  8. Babuška, I., Miller, A.: A Posteriori Error Estimates and Adaptive Techniques for the Finite Element Method. Tech. Note BN-968, Institute for Physical Science and Technology, University of Maryland 1981

  9. Babuška, I., Miller, A., Vogelius, M.: Adaptive Methods and Error Estimation for Elliptic Problems of Structural Mechanics. In: Adaptive Computational Methods for Partial Differential Equations (Babuška, I., Chandra, J., Flaherty, J.E., eds., pp. 57–73. Philadelphia: SIAM 1983

    Google Scholar 

  10. Babuška, I., Szabo, B., Katz, I.: Thep-Version of the Finite Element Method. Siam J. Numer. Anal.18, 515–545 (1981)

    Google Scholar 

  11. Babuška, I., Yu, D.: Asymptotically Exact A-Posterior Error Estimator for Biquadratic Elements. Tech. Note BN-1050, Institute for Physical Science and Technology, University of Maryland 1986

  12. Babuška, I., Yu, D.: A-posteriori Error Estimation for Biquadratic Elements and Adaptive Approaches. 1988 (to appear)

  13. Bieterman, M., Babuška, I.: The Finite Element Method for Parabolic Equations. I. A posteriori Error Estimation. Numer. Math.40, 339–371 (1982)

    Google Scholar 

  14. Bieterman, M., Babuška, I.: The Finite Element Method for Parabolic Equations, II. A posteriori Error Estimation and Adaptive Approach. Numer. Math.40, 373–406 (1982)

    Google Scholar 

  15. Kapila, A.K.: Asymptotic Treatment of Chemically Reacting Systems. Advanced Publishing Program. Boston: Pitman 1983

    Google Scholar 

  16. Petzold, L.R.: A Description of DASSL: A Differential/Algebraic System Solver. Report No. Sand. 82-8637, Sandia National Laboratory 1982

  17. Thomee, V.: Negative Norm Estimates and Superconvergence in Galerkin Methods for Parabolic Problems. Math. Comput.34, 93–113 (1980)

    Google Scholar 

  18. Zienkiewicz, O.: The Finite Element Method, 3rd Ed. London: McGraw Hill 1977

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Rensselaer Polytechnic Institute, 12180-3590, Troy, NY, USA

    Slimane Adjerid & Joseph E. Flaherty

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  1. Slimane Adjerid
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  2. Joseph E. Flaherty
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Dedicated to Prof. Ivo Babuška on the occasion of his 60th birthday

This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0156 and the U.S. Army Research Office under Contract Number DAAL 03-86-K-0112

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Adjerid, S., Flaherty, J.E. Second-order finite element approximations and a posteriori error estimation for two-dimensional parabolic systems. Numer. Math. 53, 183–198 (1988). https://doi.org/10.1007/BF01395884

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