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Partial Differential Equations and Functional Analysis: In by P. Grisvard (auth.), Jean Cea, Denise Chenais, Giuseppe

By P. Grisvard (auth.), Jean Cea, Denise Chenais, Giuseppe Geymonat, Jacques Louis Lions (eds.)

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4] Th. Apel, R. Miicke, and J. R. Whiteman, An adaptive finite element technique with a-priori mesh grading. Technical Report 9, BlCOM Institute of Computational Mathematics, 1993. [5] Th. -M. Siindig, and J. R. Whiteman, Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Technical Report 12, BICOM Institute of Computational Mathematics, 1993. To appear in Math. Meth. Appl. Sci. [6] P. Ciarlet, The finite element method for elliptic problems.

L E ]0,1] the grading parameter, Ti the distance of 11i to the edge (Ti := min( Xl,X2,X3 )En (x~ + ;),'1 x~)1/2) and some constant R > 0, we define real numbers hi (i = 1, ... 12) are fulfilled for i = 1, ... , m. 1), it is easy to construct such a mesh, see [4]. 4). In Section 3 we use the standard way for bounding the finite element error, namely the estimation of the interpolation error. 13) Elliptic Problems in Domains with Edges 23 does not hold for p = 2, but only for p > 2. That is why we restrict our consideration to problems with a right hand side f E £P(n) with p > 2.

13) Elliptic Problems in Domains with Edges 23 does not hold for p = 2, but only for p > 2. That is why we restrict our consideration to problems with a right hand side f E £P(n) with p > 2. Another task is to prove an approximation result for elements ni touching the edge, since the solution does not belong to W 2 ,p(ni ) (P> 2) there even if we would assume smooth data. 7), we get < C (h~~i3 + h3 ,i)lv; A~,p(ni)l, IIv - Iv; LP(n i ) II < c (h~/ + h~,i)lvj A~,p(ni)l. 16) with s= { I ~ _ 1 + 1.

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