An interior penalty DG method with correct and minimal averages, jumps and penalties for the miscible displacement problem of nonnegative characteristic form, and SUPG-type error estimates under low regularity, dominating Darcy velocity
Zhijie Du, Huoyuan Duan, Roger C E Tan, Yuanhong Wei
TL;DR
This work develops a novel interior penalty DG method for steady-state PDEs of nonnegative characteristic form, accommodating vanishing diffusion and low-regularity Darcy velocity typical of miscible displacement in porous media. By partitioning interelement boundaries into diffusion–diffusion and advection-related sets, the method applies the minimal necessary averages, jumps, and penalties and analyzes the resulting scheme via a rigorous consistency framework. The authors prove SUPG-type stability and derive error estimates that are robust to low velocity regularity, achieving optimal $O(h^{\ell+\frac{1}{2}})$ convergence in advection-dominated regimes and $O(h^{\min(\ell,\hat{\ell})})$ in general, with a stabilization parameter $\mathcal{D}_K(A,\mathbf{u},\gamma,h_K)$. This approach yields high-fidelity approximations for miscible displacement problems with vanishing diffusion, offering practical stability and convergence advantages on general meshes.
Abstract
An interior penalty DG method is proposed for the steady-state linear partial differential equations of nonnegative characteristic form, suitable for mixed second-order elliptic-parabolic and first-order hyperbolic equations. Due to the different natures of the elliptic, parabolic, and hyperbolic equations. In the new DG method, the averages, jumps and penalties are minimal, correctly and only imposed on the diffusion-diffusion element boundaries, in addition to the well-known upwind jumps associating with the advection velocity. For the advection-dominated problem, the penalties can be further reduced only being imposed on the diffusion-dominated subset of the diffusion-diffusion element boundaries.This is based on the novel, crucial technique about the multiple partitions of the set of the interelement boundaries into a number of subsets with respect to the diffusion and to the advection and on the consistency result we have proven. The new DG method is the first DG method and the first time that the continuity and discontinuity of the solution are correctly identified and justified of the general steady-state linear partial differential equations of nonnegative characteristic form. The new DG method and its analysis are applied to the miscible displacement problem of vanishing diffusion coefficient and of low regularity, dominating Darcy flow velocity which lives in $H(\operatorname{div};Ω)\cap \prod_{j=1}^J (H^r(D_j))^d$ for $r<1$ other than the usual assumption $(W^{1,\infty}(Ω))^d$. We prove the SUPG-type error estimates $\mathcal{O}(h^{\ell+\frac{1}{2}})$ for any element polynomial of degree $\ell\ge 1$ on generally shaped and nonconforming meshes, where the convergence order is independent of the regularity of the advection velocity. The SUPG-type error estimates obtained are new and the first time known under the low regularity of the advection velocity.