On Adaptive Eulerian-Lagrangian Method for Linear Convection-Diffusion Problems

TL;DR

This paper develops an adaptive Eulerian-Lagrangian method for linear convection-diffusion problems, providing new a posteriori error bounds and demonstrating optimal convergence and robustness through numerical tests.

Contribution

It introduces a novel a posteriori error estimation technique for ELM that estimates temporal errors along characteristics and combines it with spatial residual estimators.

Findings

Achieves optimal convergence rates for solutions with minimal regularity.

Demonstrates efficiency and robustness of the adaptive algorithm through numerical experiments.

Provides a new a posteriori error bound for ELM semi-discretization.

Abstract

In this paper, we consider the adaptive Eulerian--Lagrangian method (ELM) for linear convection-diffusion problems. Unlike the classical a posteriori error estimations, we estimate the temporal error along the characteristics and derive a new a posteriori error bound for ELM semi-discretization. With the help of this proposed error bound, we are able to show the optimal convergence rate of ELM for solutions with minimal regularity. Furthermore, by combining this error bound with a standard residual-type estimator for the spatial error, we obtain a posteriori error estimators for a fully discrete scheme. We present numerical tests to demonstrate the efficiency and robustness of our adaptive algorithm.