AMLI preconditioner for the p‐version of the FEM

From the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p ‐version finite element discretizations of elliptic boundary value problems. One ingredient of such a preconditioner is a preconditioner related to the Dirichlet problems. In the case of Poisson's equation, we present a preconditioner for the Dirichlet problems which can be interpreted as the stiffness matrix K h,k resulting from the h ‐version finite element discretization of a special degenerated problem. We construct an AMLI preconditioner C h,k for the matrix K h,k and show that the condition number of C K h,k is independent of the discretization parameter. This proof is based on the strengthened Cauchy inequality. The theoretical result is confirmed by numerical examples.