In order to obtain good results from the finite element method the mesh used should suit the behaviour of the field. The adaptive mesh generation offers an automatic way to generate meshes fitting the problem. In this paper the effectiveness of the adaptive mesh generation is measured by comparing the error in the field solution of uniformly and adaptively generated meshes both for the 2D and the 3D case.

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The resistive tearing mode is expected to be rather sensitive to a magnetic field component perpendicular to the singular layer (B⊥) that typically occurs in weakly two-dimensional sheets, and a stabilizing effect was suggested by previous authors. A normal mode analysis is performed by rigorous application of the singular perturbation technique, i.e., the scalings are determined from the significant degeneration of the differential operator and matching is accomplished by the overlap hypothesis. For B⊥/B0=O(ε2α), a tendency of the plasma flow to follow the (curved) field lines is demonstrated (ε is the width of the singular layer and α is the wavenumber of the perturbation, both are normalized to the thickness of the current sheet). The dispersion relation, however, derived in closed analytical form for S−2/5⩽α=O(1), shows no influence of the normal component on the growth rate (S is the magnetic Reynolds number). Thus, the classical results of Furth, Killen, and Rosenbluth are recovered and an analytical expression for their numerical results in the regime of the maximum growth rate is given.

Several theoretical approaches to the problem of the onset of solar flares identify the point of onset as a 'critical' point, where the solution curve turns back. In other theories, a flare occurs at a conventional transition from stability to instability. The present paper provides stability results relevant for both types of approaches. Restricting the discussion to two-dimensional configurations (e.g. models for two-ribbon flares) stability is discussed in the framework of ideal magnetohydrodynamics, resistive magnetohydrodynamics and Vlasov theory. In most (but not all) cases the perturbation modes considered are restricted to two spatial dimensions in the same sense as the equilibrium. Several rather general criteria for stability of the lower part of the solution curve ('minimal solutions') are derived. Our results provide support to the concept of flare onset at a critical point.