Theh−p version of the finite element method for elliptic equations of order2m

Summary

The problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data. It has been shown in [3, 4, 5] that the solutions of the second order and fourth order equations belong to the spacesB ¹_β where the weighted Sobolev norms of thek-th derivatives are bounded byCd ^k−l(k−l)!,k≧l, l≦2 whereC andd are constants independent ofk. In this case theh−p version of the finite element method leads to an exponential rate of convergence measured in the energy norm [6, 12, 13]. Theh−p version was implemented in the code PROBE¹ [18] and has been very successfully used in the industry.

We will discuss in this paper the generalization of these results for problems of order2m. We will show also that the exponential rate can be achieved if the exact solution belongs to the spacesB ¹_β where the weighted Sobolev norm of thek-th derivatives is bounded byCd ^k−l(k−l)!,k≧l=m+1, C andd are independent ofk. In addition, if the data is piecewise analytic, then in fact the exact solution belongs to the spacesB ^m+1_β .

Problems of this type are related obviously to many engineering problems, such as problems of plates and shells, and are also important in connection with well-known locking problems.