Finite element, unstructured mesh methods are now just beginning to be used in magnetohydrodynamics (MHD) computations. Unstructured mesh methods have become a leading approach in computational fluid dynamics for two main reasons.
First, they allow adaptive local mesh refinement. MHD motion tends to develop sharp structures: nearly discontinuous magnetic field and sharply localized, intense current sheets. We present simulations showing the formation of current sheets, with the current density increasing exponentially in time. If the resolution is inadequate, truncation error can cause artificial numerical dissipation. Therefore it is important to refine the grid as much as possible where current sheets form. This should be done adaptively, because the location of the sheets can change, as in the case of the tilt mode.
A second feature of unstructured mesh methods is their ability to fit irregular boundaries. Applications to modelling laboratory plasmas may require the use of irregularly shaped boundaries. Although this is an important feature, in this paper we emphasize adaptive mesh refinement to resolve current sheets.
In the following, we first list the incompressible, two dimensional MHD equations. We then give the standard stream function - vorticity advection form of the equations, as well as the symmetrized current - vorticity advection formulation. The equations can be discretized using piecewise linear, triangular finite elements. Three sparse matrices, the mass matrix, stiffness matrix, and bracket tensor, arise in the discretization. Their construction and assembly is discussed. The stiffness matrix can cause a convergence problem in computing the current. We give two possible cures.
The most effective cure is to use symmetrized MHD equations, in which vorticity and current are time advanced, and the potentials are found by solving Poisson equations. The other approach is to use a modified stiffness matrix with a wider stencil, having acceptable convergence properties.
Adaptive gridding is done with two mesh operations: splitting pairs of triangles into four triangles; and the inverse operation of combining four triangles into two.
After describing a new finite element code, FEMHD, based on these algorithms, we give examples of its use. We verify that the code reproduces previously known solutions for the coalescence and tilt instability. We carry out simulations of current sheet formation caused by these instabilities. We note that similar results on current sheet formation have recently been obtained using adaptive structured mesh methods [11]. This approach also used symmetrized MHD equations in the Elsässer form, to be given below.