A New Regularity Criterion for the Three-Dimensional Incompressible Magnetohydrodynamic Equations in the Besov Spaces

Regularity criteria of the weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamic (MHD) equations are discussed. Our results imply that the scalar pressure field π plays an important role in the regularity problem of MHD equations. We derive that the weak solution u , b is regular on 0 , T , which is provided for the scalar pressure field π in the Besov spaces.

Parallel two-step finite element algorithm for the stationary incompressible magnetohydrodynamic equations

Purpose The purpose of this paper is to propose a local parallel finite element algorithm based on fully overlapping domain decomposition technique to solve the incompressible magnetohydrodynamic equations. Design/methodology/approach The algorithm uses a lower-order element pair to compute an initial approximation by the Oseen-type iteration and uses a higher-order element pair to solve a linear system in each processor. Findings Besides, the convergence analysis of local parallel finite element algorithm is given. Finally, numerical experiments are presented to verify the efficiency of the proposed algorithm. Originality/value Compared with the numerical solution of the common two-step method, this method is easy to realize and can produce a more accurate solution. And, this approach is executed in parallel, so it saves a lot of computational time.