This paper focuses on some efficient deflated preconditioners for static elastic crack problems modelled by the geometrical extended finite element method. We not only construct the deflation subspace matrix which is suitable for linear elastic crack problems, but also give the principle for selecting the deflated mesh nodes. To further accelerate the convergence, we combine the deflation technique with the "crack tip" domain decomposition preconditioners through multiplicative way, and propose efficient adapted deflated preconditioned conjugate gradient solvers which can eliminate the high-frequency and low-frequency errors simultaneously in the iterations. Numerical experiments demonstrate the effectiveness of our algorithm.

This paper mainly discusses some effective domain decomposition preconditioners for static elastic crack problems modeled by geometrical extended finite element method. To construct the Schwarz type preconditioners, we adopt a special crack-tip domain decomposition strategy. The finite element mesh is decomposed into "crack tip" subdomains, which contain all the degrees of freedom (DOFs) of the branch enrichment functions, and "regular" subdomains, which contain the standard DOFs and the DOFs of the Heaviside enrichment functions. Based on the crack-tip domain decomposition strategy, an effective class of multiplicative and restrict multiplicative Schwarz preconditioners are derived. In the preconditioners, the crack tip subproblems are solved exactly and the regular subproblems are solved by some inexact solvers. Numerical experiments demonstrate the effectiveness of the preconditioners.