A Posteriori Estimator for 1D WG Transport

Abstract

This paper develops a full computable a posteriori estimator for a one-dimensional stationary transport reaction equation discretized by a global stabilization weak Galerkin method. While a continuous piecewise linear reconstruction is organized from the nodal element, the cell and nodal unknowns of the distinct solution are retained. The reconstruction rest is raised along the transport feature. In one space dimension, the raised rest is just the reconstruction mistake; accordingly, its transport energy provides an exact a posteriori identity and a crisp estimator with unit effectiveness up to numerical quadrature. A coarse residual norm upper edge and a computable cell-to-reconstruction disagreement give a guaranteed edge for the WG cell variable. Local shares of the upraised energy thrust a Dörfler-marked adaptive loop, and two smooth constructed instances affirm second order decay and unit effectiveness, while a sharp-front instance demonstrates that 19 adaptively selected elements cut the reconstruction mistake to 1.881×10⁻3, compared with 1.261×10⁻2 on a uniform mesh of the identical size. Wall time measuring’s quantify the additional cost of residual raising, and the share is an error-certification and adaptive-control mechanism for a generic WG discretization instead than a novel inflated strategy.