SUMMARYWe present two efficient methods of two-grid scheme for the approximation of two-dimensional semilinear reaction-diffusion equations using an expanded mixed finite element method. To linearize the discretized equations, we use two Newton iterations on the fine grid in our methods. Firstly, we solve an original non-linear problem on the coarse grid. Then we use twice Newton iterations on the fine grid in our first method, and while in second method we make a correction on the coarse grid between two Newton iterations on the fine grid. These two-grid ideas are from Xu's work (SIAM J. Sci. Comput. 1994; 15:231-237; SIAM J. Numer. Anal. 1996; 33:1759-1777) on standard finite element method. We extend the ideas to the mixed finite element method. Moreover, we obtain the error estimates for two algorithms of two-grid method. It is showed that coarse space can be extremely coarse and we achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O(h 1/4 ) in the first algorithm and H = O(h 1/6 ) in second algorithm.

In this paper, we consider the problem of the steady-state fully developed magnetohydrodynamic (MHD) flow of a conducting fluid through a channel with arbitrary wall conductivity in the presence of a transverse external magnetic field with various inclined angles. The coupled governing equations for both axial velocity and induced magnetic field are firstly transformed into decoupled Poisson-type equations with coupled boundary conditions. Then the dual reciprocity boundary element method (DRBEM) [20] is used to solve the Poisson-type equations. As testing examples, flows in channels of three different crosssections, rectangular, circular and triangular, are calculated. It is shown that solutions obtained by the DRBEM with constant elements are accurate for Hartmann number up to 8 and for large conductivity parameters comparing to exact solutions and solutions by the finite element method (FEM).

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