Two-level finite element method with a stabilizing subgrid for the incompressible MHD equations

We consider the Galerkin finite element method (FEM) for the incompressible magnetohydrodynamic (MHD) equations in two dimension. The domain is discretized into a set of regular triangular elements and the finite-dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual-free bubble functions. To find the bubble part of the solution, a two-level FEM with a stabilizing subgrid of a single node is described and its application to the MHD equations is displayed. Numerical approximations employing the proposed algorithm are presented for three benchmark problems including the MHD cavity flow and the MHD flow over a step. The results show that the proper choice of the subgrid node is crucial to get stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. Furthermore, the approximate Solutions obtained show the well-known characteristics of the MHD flow. Copyright (C) 2009 John Wiley & Sons, Ltd

A monolithic approach for the incompressible magnetohydrodynamics equations

A numerical algorithm has been developed to solve the incompressible magnetohydrodynamics (MHD) equations in a fully coupled form. The numerical approach is based on the side centered finite volume approximation where the velocity and magnetic filed vector components are defined at the center of edges/faces, meanwhile the pressure term is defined at the element centroid. In order to enforce a divergence free magnetic field, a magnetic pressure is introduced to the induction equation. The resulting large-scale algebraic linear equations are solved using a one-level restricted additive Schwarz preconditioner with a block-incomplete factorization within each partitioned sub-domains. The parallel implementation of the present fully coupled unstructured MHD solver is based on the PETSc library for improving the effi- ciency of the parallel algorithm. The numerical algorithm is validated for 2D lid-driven cavity flows and backward step problems for both conducting and insulating walls

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the ow of an electrically charged fuid under the in uence of an external electromagnetic eld with thermal coupling. This system of partial di erential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the di erent physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task which requires e cient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure, current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 2 block matrix, and consider a LU preconditioner obtained by approximating the corresponding Schur complement. For every one of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknown, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a fexible and general software design for the code implementation of this type of preconditioners.Preprin