Solution techniques for the p‐version of the adaptive finite element method

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Abstract
This paper investigates the performance of a class of methods based on the Preconditioned Conjugate Gradient (PCG) method for the solution of large and sparse systems of linear equations which arise from the p‐version of the adaptive finite element method. The hierarchical type of preconditioning used in the past is extended to the SSOR and incomplete factorization type preconditioning and applied with a global handling of the stiffness matrix. The paper also presents some new ideas on Domain Decomposition (DD) matrix‐handling techniques, whereby the explicit formulation of the resulting Schur complement is avoided by utilizing the PCG method with efficient preconditioners and by processing separately on the coarse and the fine mesh. The suitability of the methods to the p‐version of the adaptive finite element method is demonstrated through extensive numerical tests in two dimensions.

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