Domain decomposition preconditioners for thin rectangular P-version finite elements

The use of the p-version finite element method to solve an elliptic partial differential equation results in a very large linear system which must be solved numerically. We consider iterative solution, using the conjugate gradients method, preconditioned by domain decomposition preconditioners, and analyze the condition number of two recently developed such preconditioners for problems in ${\cal R}\sp2$. The p-version finite element method achieves accuracy by increasing the degree p of the elements, rather than by decreasing their diameter, as in the more familiar h-version. Thus, there are many degrees of freedom on each element. The preconditioners considered here treat each element as a subdomain, allowing much of the work of generating and solving the preconditioner to be performed on each element independently. Most published analyses of preconditioners for the finite element method do not consider the case of elements with high aspect ratios, a case which happens often in practice, and consider only elements which are nearly, e.g. square. We show that, for the first preconditioner considered here, the condition number grows with the square of the aspect ratio, and thus its efficacy degrades significantly. We also show that, for the second preconditioner, which adapts to elements of different aspect ratios, the condition number is bounded independently of the aspect ratio. Thus the second preconditioner can be more effective than the first for the case of very thin elements. We accomplish this analysis using energy seminorms which are defined especially for thin subdomains, and basis polynomials which are orthogonal with respect to these seminorms. This approach gives a greater insight into the behaviour of these methods than previously published methods.