Error Estimates for First- and Second-Order Lagrange-Galerkin Moving Mesh Schemes for the One-Dimensional Convection-Diffusion Equation

TL;DR

The paper tackles accurate numerical simulation of the one-dimensional convection–diffusion equation in convection-dominated regimes by introducing the Lagrange–Galerkin Moving Mesh (LGMM) framework, which couples LG time stepping with a dynamically evolving mesh. It provides mass-conserving first- and second-order LG schemes and proves optimal error estimates in the combined ℓ∞(L2) and ℓ2(H1) norm, alongside new time-dependent interpolation bounds. The analysis shows that the moving mesh preserves mass, remains stable, and achieves convergence rates of O(Δt + h^2) for the first-order scheme and O(Δt^2 + h^2) for the second-order scheme on linear elements. Numerical experiments confirm the theoretical results and demonstrate that LGMM can suppress oscillations and better resolve sharp concentration features compared to fixed-mesh LG methods. The framework is readily extensible to higher dimensions and holds promise for applications in biology and fluid dynamics where mass preservation and adaptive meshing are beneficial.

Abstract

A new moving mesh scheme based on the Lagrange-Galerkin method for the approximation of the one-dimensional convection-diffusion equation is studied. The mesh movement, which is prescribed by a discretized dynamical system for the nodal points, follows the direction of convection. It is shown that under a restriction of the time increment the mesh movement cannot lead to an overlap of the elements and therefore an invalid mesh. For the linear element, optimal error estimates in the $\ell^\infty(L^2) \cap \ell^2(H_0^1)$ norm are proved in case of both, a first-order backward Euler method and a second-order two-step method in time. These results are based on new estimates of the time dependent interpolation operator derived in this work. Preservation of the total mass is verified for both choices of the time discretization. Numerical experiments are presented that confirm the error estimates and demonstrate that the proposed moving mesh scheme can circumvent limitations that the Lagrange-Galerkin method on a fixed mesh exhibits.

Figures (3)

• Figure 1: Numerical solution $\phi_h$ and corresponding mesh sizes in Example \ref{['ex2']} over the computational domain at time instances $t = 0$ (left), $t=0.2340$ (center) and $t=0.4875$ (right) obtained by the LGMM scheme for $\nu=0.01$ and $N=512$.
• Figure 2: Numerical solution $\phi_h$ and corresponding mesh sizes in Example \ref{['ex1']} over the computational domain at time instances $t = 0$ (left), $t=1$ (center) and $t=2$ (right) obtained by the LG scheme with fixed mesh ($N=256$). The numerical solution exhibits oscillations.
• Figure 3: Numerical solution $\phi_h$ and corresponding mesh sizes in Example \ref{['ex1']} over the computational domain at time instances $t = 0$ (left), $t=1$ (center) and $t=2$ (right) obtained by the LGMM scheme with $N=256$. The nodal points aggregate along with the solution $\phi_h$.

Theorems & Definitions (27)

• Definition 1
• Remark 1
• Remark 2
• Remark 3
• Theorem 1: Non-overlapping condition for the moving mesh method
• Remark 4
• Theorem 2
• Remark 5
• Corollary 1
• Corollary 2: Mass preserving property of the first-order LGMM Scheme