Sharp 𝐿¹ a posteriori error analysis for nonlinear convection-diffusion problems

Abstract

Abstract. We derive sharp L ∞ (L 1 ) a posteriori error estimates for initial boundary value problems of nonlinear convection-diffusion equations of the form ∂u ∂tunder the nondegeneracy assumption A (s) > 0 for any s ∈ R. The problem displays both parabolic and hyperbolic behavior in a way that depends on the solution itself. It is discretized implicitly in time via the method of characteristic and in space via continuous piecewise linear finite elements. The analysis is based on the Kružkov "doubling of variables" device and the recently introduced "boundary layer sequence" technique to derive the entropy error inequality on bounded domains. The derived a posteriori error estimators have the correct convergence order in the region where the solution is smooth and recover the standard a posteriori error estimators known for parabolic equations with strong diffusions.

Effects of Integrations and Adaptivity for the Eulerian--Lagrangian Method

“…We use the so-called guaranteed error reduction strategy in [25] and the spatial error estimators proposed in [17]. Our implementation is based on a Matlab adaptive finite element package, AFEM@matlab [26].…”

“…A posteriori error analysis and spatial mesh adaptivity have been applied to the ELM; see [14][15][16][17]. In particular, [16] gave a residual-based L^{L'^) a posteriori error estimator; however, the numerical experiments therein indicated that the norm of the residual on an individual element may be a poor estimate of the local error (the norm of the residual can be used to bound the error on a global basis from above.)…”

“…In particular, [16] gave a residual-based L^{L'^) a posteriori error estimator; however, the numerical experiments therein indicated that the norm of the residual on an individual element may be a poor estimate of the local error (the norm of the residual can be used to bound the error on a global basis from above.) In [17], the authors derived a sharp L°°{L^) a posteriori error estimator for a nonlinear convection-diffusion equation, which is discretized with the ELM implicitly in time and the continuous piecewise linear finite element in space. We will use the spatial error estimators proposed in [17] to drive out adaptive mesh refinements in Sections 5 and 6.…”

“…In [17], the authors derived a sharp L°°{L^) a posteriori error estimator for a nonlinear convection-diffusion equation, which is discretized with the ELM implicitly in time and the continuous piecewise linear finite element in space. We will use the spatial error estimators proposed in [17] to drive out adaptive mesh refinements in Sections 5 and 6.…”

Lifting solutions of quasilinear convection-dominated problems

“…These unphysical behaviours can be eliminated considering several approaches (see, for instance [7][8][9][10]14,[16][17][18]22]. The numerical difficulties due to the lack of smoothness when Newton's method is used to solve the algebraic system resulting from the discretization are considered, for example, in [2,4,11,12].…”