A stable extrapolation method for multidimensional degenerate parabolic problems

Abstract:

Degenerate parabolic problems in several space variables are approximated by combining a preliminary regularization procedure with a finite element extrapolation method. The proposed extrapolation acts on the so-called phase variable and leads to a linear problem which is shown to be stable. The ensuing linear algebraic system involves the same matrix for all time steps. Energy error estimates are also derived for the physical unknowns. An $O({h^{1/2}})$ rate of convergence is proved, provided the approximation parameters are suitably related. In case the linear systems are solved by an iterative algorithm, such as the conjugate gradient method, an $O({h^{3/2}})$ tolerance for the error reduction is shown to preserve the overall accuracy; the required computational effort is thus nearly optimal.