Optimal Energy-Norm Convergence for the Dynamic Diffusion Finite Element Method

Abstract

We revisit the nonlinear two-scale Dynamic Diffusion (DD) finite element formulation mathematically analyzed by Santos et al. (2021) for stationary advection--diffusion--reaction problems. We establish an explicit local Lipschitz estimate for the artificial diffusivity, with a constant of order $h_T$, and use it to prove uniqueness of the discrete solution for sufficiently fine meshes. By separating the approximation error in the energy norm from the contribution associated with artificial diffusion, we derive an optimal first-order a priori energy-norm estimate for continuous piecewise linear finite elements enriched with simplex bubble functions. We further prove that the square root of the nonlinear artificial dissipation is $O(h)$. Consequently, the combined energy-error and artificial-dissipation measure also converges with first order, sharpening the previously available $O(h^{1/2})$ estimate. A smooth manufactured problem, considered with and without reaction, corroborates the predicted convergence rates. A second, strongly advection-dominated problem with sharp outflow layers illustrates the stabilizing behavior of the DD formulation.