Theh−p version of the finite element method for elliptic equations of order2m
Theh−p version of the finite element method for elliptic equations of order2m
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DOI：
10.1007/bf01395885
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发表时间：
1988
影响因子：
2.1
通讯作者：
B. Guo
中科院分区：
文献类型：
--
作者：
B. Guo
工程问题中的椭圆型偏微分方程问题通常用分段解析数据来刻画。文[3，4，5]证明了二阶和四阶方程的解属于空间B β 1，其中k阶导数的加权Sobolev范数有界于Cdk − 1（k− 1）！，k ≠ l，l ≠ 2其中坎德尔常数与k无关。在这种情况下，有限元方法的h-p版本会导致以能量范数测量的指数收敛率[6，12，13]。h-pversion在程序PROBE 1 [18]中实现，并在工业中得到了非常成功的应用。本文将讨论这些结果对2 m阶问题的推广。我们还将证明，如果精确解属于空间B β 1，其中k阶导数的加权Sobolev范数由Cdk − 1（k− 1）！，k ≠ l=m+1，坎德尔与k无关。此外，如果数据是分段解析的，则实际上精确解属于空间B βm+1。这类问题显然与许多工程问题有关，如板壳问题，而且与著名的锁定问题有关。
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 spacesBβ1where the weighted Sobolev norms of thek-th derivatives are bounded byCdk−l(k−l)!,k≧l, l≦2whereCanddare constants independent ofk. In this case theh−pversion of the finite element method leads to an exponential rate of convergence measured in the energy norm [6, 12, 13]. Theh−pversion 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 spacesBβ1where the weighted Sobolev norm of thek-th derivatives is bounded byCdk−l(k−l)!,k≧l=m+1, Canddare independent ofk. In addition, if the data is piecewise analytic, then in fact the exact solution belongs to the spacesBβm+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.