Preconditioning on element interfaces for the p-version finite element method and spectral element method

In the preconditioned iterative method based on nonoverlapping domain decompositions, the preconditioner is usually of block diagonal type. To calculate its action on a residual vector, it is required to solve a linear system of algebraic equations with the coefficient matrix being the Schur complement block corresponding to the interface. To reduce the computational cost, the Schur complement blocks in the preconditioners can be replaced with other simpler matrices, which are called the interface preconditioners. In this paper, we discuss various interface preconditioners for the p-version finite element method and spectral element method in R-2 and R-3. Emphases are placed on (1) the lower and upper bounds of the condition numbers for the preconditioned Schur complement blocks using the interface preconditioners, which characterize the enlargement of the overall condition numbers, and (2) the algorithms to efficiently implement the interface preconditioners. Numerical results are presented for the condition numbers using various interface preconditioners.