The h-p version of the finite element method for elliptic equations of order 2m (Q1093334)

The problems of elliptic partial differential equations stemming from engineering problems are usually characterized by piecewise analytic data. It has been shown that the solutions of the second order and fourth order equations belong to the countable normed spaced \(B^{\ell}_{\beta}\) where the weighted Sobolev norms of the k-th derivatives are bounded by \(Cd^{k-\ell}(k-\ell)!\), \(k\geq \ell\), \(\ell =2,3\) where C and d are constants independent of k. In this case the h-p version of the finite element method leads to an exponential rate of convergence measured in the energy norm. The h-p version was implemented in the code PROBE (NOETIC TECHNOLOGIES, Inc., St. Louis, MO) and has been very successfully used in the industry. We discuss the generalization of these results for problems of order 2m. We show also that the exponential rate can be achieved if the exact solution belongs to the countable normed spaces \(B^{\ell}_{\beta}\) where the weighted Sobolev norm of the k-th derivatives is bounded by \(Cd^{k-\ell}(k-\ell)!\), \(k\geq \ell =m+1\), C and d are independent of k. In addition, if the data is piecewise analytic, then in fact the exact solution belongs to the spaces \(B_{\beta}^{m+1}\). 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.