A modified Hermitian and skew-Hermitian preconditioner for the Ohta-Kawasaki equation
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In this paper, block preconditioners for the discretized Ohta-Kawasaki partial differential equation are proposed. We first discretize the Ohta-Kawasaki dynamic equation by time discretization via the convex splitting scheme and spatial discretization via the finite element method. The time discretized scheme is unconditionally energy stable. Due to the ill-conditional feature of the discretized linear system, the preconditioning approaches are required. Then, based on Schur complement, using the construction technique in the recent results, we propose block triangular preconditioners and give the spectral distribution for the preconditioning system. Further, by arranging the discretization system to a generalized saddle point problem, we offer a modified Hermitian and skew-Hermitian splitting (MHSS) block preconditioner for the discretized Ohta-Kawasaki partial differential equation. The distribution of the eigenvalues for the preconditioned matrix is analyzed. Moreover, we present an adaptive m-step polynomial preconditioner to approximate the inverse for the (1, 1) position block matrix in the preconditioned matrix to get better effect of computing the block MHSS preconditioner. Finally, numerical examples are given to illustrate the effectiveness of the proposed preconditioners for the Ohta-Kawasaki equation.
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