Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

Abstract

SUMMARYThe paper introduces a methodology to compute strict upper and lower bounds for linear-functional outputs of the exact solutions of the advection-reaction-diffusion equation. The bounds are computed using implicit a-posteriori error estimators from stabilized finite element approximations of the exact solution. A new methodology is introduced, based in the ideas presented in [1] for the Galerkin formulation, that allows obtaining bounds also for stabilized formulations. This methodology is combined with both hybrid-flux and flux-free techniques for error assessment. The application to stabilized formulations provides sharper estimates than when applied to Galerkin methods. The best results are found in combination with the fluxfree technique.

Goal-oriented space-time adaptivity for transient dynamics using a modal description of the adjoint solution

“…Similarly as when splitting the space and time contributions, criteria (26) are stronger than (25). This is more relevant for large values of N bk , because the effect of the triangular inequalities in the equations (27) is more important.…”

“…Thus, the adapted numerical solution might be very conservative if the number of blocks N bk is large. An additional condition is added to (26) in order to allow unrefinement (mesh coarsening). Note that the conditions (26) indicate only if the solution is acceptable and, if not, if the mesh has to be refined.…”

“…An additional condition is added to (26) in order to allow unrefinement (mesh coarsening). Note that the conditions (26) indicate only if the solution is acceptable and, if not, if the mesh has to be refined. They do not provide a criterion to unrefine the discretization when the error indicators η s m and η t m are small enough.…”

“…Moreover, checking for unrefining at each adaptive step may result in an unstable scheme. As previously said, conditions (26) and (28) are the criteria allowing to decide if the numerical solution is accepted or rejected inside the interval I bk m . If conditions (26) and (28) hold (or only (26) after the first adaptive iteration), then the solution is accepted.…”

“…For instance, for elliptic problems [20-22, 29, 19], for the convection-diffusionreaction equation [25,26], for non-linear structural problems [16,17], for time-dependent parabolic problems [23,24,5] and for elastodynamics (or other 2nd order hyperbolic problems) [2][3][4]11].…”