A genuinely stable Lagrange-Galerkin scheme for convection-diffusion problems
This work provides a provably stable numerical method for convection-diffusion problems, addressing instability from quadrature errors in conventional schemes.
The authors developed a Lagrange-Galerkin scheme for convection-diffusion problems that avoids numerical quadrature, ensuring genuine stability. They proved stability and optimal-order convergence, with numerical results confirming the theory.
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.