A genuinely stable Lagrange-Galerkin scheme for convection-diffusion problems

TL;DR

This paper introduces a stable Lagrange-Galerkin scheme for convection-diffusion problems that avoids numerical quadrature errors, ensuring theoretical stability and convergence with practical numerical validation.

Contribution

The paper proposes a genuinely stable Lagrange-Galerkin scheme that can be implemented exactly, eliminating quadrature-induced instability in convection-diffusion simulations.

Findings

The scheme is proven to be stable and convergent of the best possible order.

Numerical results confirm the theoretical stability and convergence estimates.

Abstract

We present a Lagrange-Galerkin scheme free from numerical quadrature for convection-diffusion problems. Since the scheme can be implemented exactly as it is, theoretical stability result is assured. While conventional Lagrange-Galerkin schemes may encounter the instability caused by numerical quadrature error, the present scheme is genuinely stable. We prove the stability and convergence of the best possible order. Numerical results reflect these estimates.