Residual-Based a Posteriori Error Estimators for Algebraic Stabilizations

TL;DR

The paper addresses residual-based a posteriori error estimation for algebraic stabilization schemes in convection-diffusion-reaction problems, extending the AFC-based estimator from Jha20 to MUAS/SMUAS. It develops an energy-norm estimator with explicit components $\eta_1$, $\eta_2$, and $\eta_3$, and shows a global bound $\|u-u_h\|_a^2 \le \eta^2$, incorporating element residuals, face residuals, and stabilization terms. Numerical tests in 2D on adaptively refined grids compare SMUAS with AFC (BJK limiter), finding comparable accuracy but significantly higher efficiency for SMUAS (fewer nonlinear iterations and rejections). The results support applying residual-based adaptivity to algebraically stabilized FEM on convection-dominated problems, enabling reliable refinement and efficient stabilization in practical computations.

Abstract

In this note, we extend the analysis for the residual-based a posteriori error estimators in the energy norm defined for the algebraic flux correction (AFC) schemes [Jha20.CAMWA] to the newly proposed algebraic stabilization schemes [JK21.NM, Kn23.NA]. Numerical simulations on adaptively refined grids are performed in two dimensions showing the higher efficiency of an algebraic stabilization with similar accuracy compared with an AFC scheme.

Figures (2)

• Figure 1: Error in the energy norm (left), the global upper bound estimator $\eta$ defined in Eq. \ref{['eq:post_upper_bound']} (center), and the number of nonlinear iterations and rejections (right). In the right figure, the left $y$-axis corresponds to the AFC scheme with the BJK limiter and the right $y$-axis corresponds to the SMUAS method.
• Figure 2: Adaptively refined grids with $\#\mathrm{dofs}\approx 25,000$. AFC scheme with BJK limiter (left) and SMUAS method (right).

Theorems & Definitions (1)

• Remark 1