The breakdown of a positive point to plane gap in air near atmospheric pressure begins with the formation of a low-conductivity filament by the space-charge-controlled streamer process. Within the filament, the rate of electron attachment exceeds that of ionization, and the external current decreases. However, a sudden rise of current leading to spark breakdown can be observed. The following explanation is proposed. Owing to the current flow the temperature increases within the discharge. A radial flow of neutral species will thus appear, which decreases the neutral density near the discharge axis. In turn E/N increases until the ionization rate becomes greater than that of attachment, leading to the final current growth.

We consider the Galerkin method to solve a parabolic initial boundary value problem in one space variable, using piecewise polynomial functions and give an alternative proof of superconvergence. Then by means of Lobatto quadrature, we obtain purely explicit vector initial value problems without loss in the order of accuracy, global or pointwise.

Abstract. In the case of one-dimensional Galerkin methods the phenomenon of superconvergence at the knots has been known for years [5], [7]. In this paper, a minor kind of superconvergence at specific points inside the segments of the partition is discussed for two classes of Galerkin methods: the Ritz-Galer kin method for 2mth order self-adjoint boundary problems and the collocation method for arbitrary mth order boundary problems. These interior points are the zeros of the Jacobi polynomial P';:·m (

As is known [41, the C ~ Galerkin solution of a two-point boundary problem using piecewise polynomial functions, has O(h 2k) convergence at the knots, where k is the degree of the finite element space. Also, it can be proved [5] that at specific interior points, the Gauss-Legendre points the gradient has O(h k+ 2) convergence, instead of O(h~). In this note, it is proved that on any segment there are k-1 interior points where the Galerkin solution is of o(hk+2), one order better than the global order of convergence. These points are the Lobatto points. Subject Classifications: AMS (MOS) 65 N 30; CR: 5.17.