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A constructive method for deriving finite elements of nodal type

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Summary

In this paper, we propose an algorithm to derive nodal methods corresponding to various two and three-dimensional nonconforming and mixed finite elements. We show that this algorithm can be used to obtain several classical schemes as well as some more recently developed schemes, and that it leads to a simple proof of unisolvence for these methods. Finally we use our method to obtain a three dimensional nodal scheme of BDM type.

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References

  1. Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. M2AN,19, 7–32 (1985)

    Google Scholar 

  2. Bramble, J.H., Hilbert, S.R.: Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal.7, 113–124 (1970)

    Google Scholar 

  3. Brezzi, F., Douglas, J. Jr., Duran, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math.51, 237–250 (1987)

    Google Scholar 

  4. Brezzi, F., Douglas, J. Jr., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math.47, 217–235 (1985)

    Google Scholar 

  5. Chavent, G., Cohen, G., Jaffre, J.: Discontinuous upwinding and mixed finite elements for two-phase flow in reservoir simulation. Comput. Methods Appl. Mech. Eng.47, 93–118 (1984)

    Google Scholar 

  6. Chavent, G., Jaffre, J.: Mathematical models and finite elements in reservoir simulation. Amsterdam: North Holland 1986

    Google Scholar 

  7. Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam: North Holland 1978

    Google Scholar 

  8. Del Valle, E., Hennart, J. P., Meade, D.: Finite element formulations of nodal schemes for neutron diffusion and transport problems. Nucl. Sci. Eng.92, 204–211 (1986)

    Google Scholar 

  9. Dorning, J.: Modern coarse-mesh methods—A development of the ‘70’s. Comput. Methods Nucl. Eng. vol. 1, pp. 3.1–3.31. American Nuclear Society, Williamsburg, Virginia 1979

    Google Scholar 

  10. Douglas, J. Jr.: Numerical methods for the flow of miscible fluids in porous media. In: Lewis, R.W., Bettess, P., Hinton, E. (eds.) Numerical methods in coupled systems, pp. 405–439. New York: Wiley 1984

    Google Scholar 

  11. Fedon-Magnaud, C., Hennart, J.P., Lautard, J.J.: On the relationship between some nodal schemes and the finite element method in static diffusion calculations. Adv. Reactor Comput. vol. 2, pp. 987–1000. American Nuclear Society, Salt Lake City, Utah 1983

    Google Scholar 

  12. Finnemann, H., Bennewitz, F., Wagner, M.F.: Interface current techniques for multidimensional reactor calculations. Atomkernenergie30, 123–128 (1977)

    Google Scholar 

  13. Fröhlich, R.: Summary discussion and state of the art review for coarse-mesh computational methods. Atomkernenergie30, 152–158 (1977)

    Google Scholar 

  14. Gladwell, I., Wait, R.: A survey of numerical methods for partial differential equations. Oxford: Clarendon 1979

    Google Scholar 

  15. Hennart, J.P.: A general approach to nodal schemes in numerical transport theory. Com. Técnicas, Serie Naranja: No 382, 24 p., IIMAS-UNAM 1985

  16. Hennart, J.P.: Nodal schemes, mixed-hybrid finite elements and block-centered finite differences. Rapport de Recherche No 386, 59, p., INRIA 1985

  17. Hennart, J.P.: A general family of nodal schemes. SIAM J. Sci. Stat. Comput.3, 264–287 (1986)

    Google Scholar 

  18. Hennart, J.P.: A general finite element framework for nodal methods. In: Whiteman, J.R. (ed.) MAFELAP 1984, pp. 309–316. London: Academic Press 1986

    Google Scholar 

  19. Langenbuch, S., Maurer, W., Werner, W.: Coarse-mesh flux-expansion method for the analysis of space-time effects in large light water reactors cores. Nucl. Sci. Eng.63, 437–456 (1977)

    Google Scholar 

  20. Langenbuch, S., Maurer, W., Werner, W.: High-order schemes for neutron kinetics calculations, based on a local polynomial approximation. Nucl. Sci. Eng.64, 508–516 (1977)

    Google Scholar 

  21. Lesaint, P.: On the convergence of Wilson's nonconforming element for solving the elastic problem. Comput. Methods Appl. Mech. Eng.7, 1–16 (1976)

    Google Scholar 

  22. Nedelec, J.C.: Mixed finite elements in ℝ3. Numer. Math.35, 315–341 (1980)

    Google Scholar 

  23. Nedelec, J.C.: A new family of mixed finite element in ℝ3. Numer. Math. (To appear)

  24. Nitsche, J.A.: Convergence of nonconforming methods. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 15–53. New York: Academic Press 1974

    Google Scholar 

  25. Raviart, P.A., Thomas, J.M.: A mixed finite element method for second order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of the Finite Element Methods. (Lecture Notes in Mathematics 606, pp. 292–315.) Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  26. Shober, R.A., Sims, R.N., Henry, A.F.: Two nodal methods for solving time-dependent group diffusion equations. Nucl. Sci. Eng.64, 582–592 (1977)

    Google Scholar 

  27. Strang, G., Fix, G.J.: An analysis of the finite element method. Englewood Cliffs, New Jersey: Prentice Hall 1973

    Google Scholar 

  28. Wagner, M.R., Koebke, K.: Progress in nodal reactor analysis. Advances in Reactor Computations, Vol. 2, pp. 941–962. American Nuclear Society, Salt Lake City, Utah 1983

    Google Scholar 

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Hennart, JP., Jaffre, J. & Roberts, J.E. A constructive method for deriving finite elements of nodal type. Numer. Math. 53, 701–738 (1988). https://doi.org/10.1007/BF01397137

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