A posteriori error estimation and adaptivity for degenerate parabolic problems

Abstract

Abstract. Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical space-time discretization consisting of C 0 piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method.

A posteriori error analysis for higher order dissipative methods for evolution problems

“…Most of the work has been done for linear or nonlinear dissipative problems by considering time discretizations based on the backward Euler method or on higher order discontinuous Galerkin methods, cf. e.g., [9,7,8,17,27,28] and [15,16,18].…”

“…In contrast to the approach of [7,8,9,10,11,15,16,17,28], which is based on the strong stability of suitable dual problems, the key novel ingredient of our approach to a posteriori error analysis is a higher order reconstruction U , of degree q + 1, which yields the differential equation (…”

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

“…In nondegenerate cases, Verfürth [46,47] was able to obtain an estimator which is both reliable and efficient. A pioneering contribution for degenerate parabolic problems has been obtained by Nochetto et al in [36]. Therein, L ∞ (0, T ; H −1 (Ω)) estimates for the error in the enthalpy and L 2 (0, T ; L 2 (Ω)) estimates for the error in the temperature are obtained.…”

A posteriorierror estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations

“…In [38] an a posteriori error estimate for a characteristic Galerkin approximation of linear unstationary convection-diffusion equations is deduced. Further results on a posteriori error estimates for finite element approximations were obtained in [39] for degenerate parabolic equations including the classical Stefan problem. In [2] linear stationary convection dominated anisotropic diffusion problems are considered and error estimates in the energy norm are derived.…”