Abstract
Estimates of the discretization error in the fixed membrane problem appear to be considerably more difficult to obtain than for the Diri-chlet problem. For example, the detailed estimates by Saulev 6 will illustrate this point. I shall give here a particular discrete analogue which is a natural one to consider and yet one whose error analysis is easily related to that for the Dirichlet problem. At the same time we shall considerably reduce the usual regularity assumptions in our error analyses. The results presented here are drawn, for the most part, from a paper 2 with J. H. Bramble.
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References
J. Brabmle : (C.I.M.E, Lectures, this volume)
J. Bramble and B. Hubbard : Effects of boundary regularity on the discretization error in the fixed membrane eigenvalue problem. (To appear)
Gunter : Die Potentialtheorie und ihre Anwendung auf Grundlagen der Mathematischen Physik B. G. B.G. Teubner, Leipzig (1957).
P.Laasonen: On the degree of convergence of discrete approximations for the solutions of the Dirichlet problem. Ann. Acad. S i. Fenn. Series A. 246, 1–19 (1957).
S.Lehman : Developments at an analytic corner of solutions of partial differential equations, J. Math. Mech. 8, 727–760 (1959).
V. K. Saulev : On the solution of the problem of eigenvalues by the method of finite differences, Translation in English in A. M.S. Translations (2) Vol. 8,277–287 (1958).
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Hubbard, B.E. (2010). Error Estimates in the Fixed Membrane Problem. In: Lions, J.L. (eds) Numerical Analysis of Partial Differential Equations. C.I.M.E. Summer Schools, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11057-3_8
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DOI: https://doi.org/10.1007/978-3-642-11057-3_8
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