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SUMMARYThe problem of generating a matrix A with speciÿed eigen-pair, where A is a symmetric and antipersymmetric matrix, is presented. An existence theorem is given and proved. A general expression of such a matrix is provided. We denote the set of such matrices by SAS n E . The optimal approximation problem associated with SAS n E is discussed, that is: to ÿnd the nearest matrix to a given matrix A * by A ∈ SAS n E . The existence and uniqueness of the optimal approximation problem is proved and the expression is provided for this nearest matrix.

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The problem of generating a matrix A with specified eigenpair, where A is an anti-symmetric and persymmetric matrix, is presented. The solvability conditions are studied. A general expression of such a matrix is provided. We denote the set of such matrices by AS n E . The best approximation problem associated with AS n E is discussed, that is: to find the nearest matrix to a given matrix A * by A ∈ AS n E . The existence and uniqueness of the best approximation problem is proved and the expression of this nearest matrix is provided.

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We investigate an Oseen two-level stabilized finite-element method based on the local pressure projection for the 2D/3D steady Navier-Stokes equations by the lowest order conforming finite-element pairs (i.e.,Q1−P0andP1−P0). Firstly, in contrast to other stabilized methods, they are parameter free, no calculation of higher-order derivatives and edge-based data structures, implemented at the element level with minimal cost. In addition, the Oseen two-level stabilized method involves solving one small nonlinear Navier-Stokes problem on the coarse mesh with mesh sizeH, a large general Stokes equation on the fine mesh with mesh sizeh=O(H)2. The Oseen two-level stabilized finite-element method provides an approximate solution (uh,ph) with the convergence rate of the same order as the usual stabilized finite-element solutions, which involves solving a large Navier-Stokes problem on a fine mesh with mesh sizeh. Therefore, the method presented in this paper can save a large amount of computational time. Finally, numerical tests confirm the theoretical results. Conclusion can be drawn that the Oseen two-level stabilized finite-element method is simple and efficient for solving the 2D/3D steady Navier-Stokes equations.

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