A posteriori error estimation and adaptivity for degenerate parabolic problems

R. H. Nochetto, A. Schmidt, C. Verdi
1999 Mathematics of Computation  
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 di usion coe cient. They are applied to a practical space-time discretization consisting of C 0 piecewise linear nite elements over highly graded unstructured meshes,
more » ... d backward nite di erences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an e cient adaptive algorithm, which equidistributes space and time discretization errors via re nement/coarsening. A simulation nally 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. 1991 Mathematics Subject Classi cation. 65N15, 65N30, 65N50, 80A22, 35K65, 35R35. compatible consecutive meshes. These issues are fully discussed in x6. We conclude in x7 with simulations illustrating the viability of our Approaches I and II, as well as a heuristic Approach III based on using local regularity of in (1.4) and heat estimators away from discrete interfaces. We clearly show that they are all able to detect the presence of interfaces, and re ne accordingly, and that Approaches II and III perform best. There is no need to compute the interface explicitly for mesh design, which is a major improvement with respect to 12]. Further simulations, comparisons of several nonlinear solvers, and a detailed description of the adaptive algorithm will be presented elsewhere 14]. Even though our error estimates (1.5) are rigorous, they do not necessarily imply E(u 0 ; f; T; ; U; h; ) ! 0 as h; ! 0 because E depends on discrete quantities that change with h and . Stability and convergence will be assessed in 13].
doi:10.1090/s0025-5718-99-01097-2 fatcat:hsvlwlbggvcjneudiuk3e756mi