Iterative solvers by substructuring for the p -version finite element method

We study a class of substructuring methods well-suited for iterative solution of large systems of linear equations arising from the p-version finite element method. The p-version offers a natural decomposition with every element treated as a substructure. We use the preconditioned conjugate gradient method with preconditioning constructed by a decomposition of the local function space on each element. We develop an elementary theory giving bounds on the condition numbers which do not depend on the number of elements if a sparse system with only few variables per element is solved in each iteration. This bound can be evaluated considering one element at a time and we compute such condition numbers numerically for various elements.

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