Recovery of the Interface Velocity for the Incompressible Flow in Enhanced Velocity Mixed Finite Element Method

TL;DR

This work tackles the challenge of accurate interface velocity in incompressible Darcy flow solved with Enhanced Velocity MFEM on nonmatching multiblock grids. It introduces a postprocessing strategy that recovers a smoother pressure $\tilde{p}_h$ and employs Oswald interpolation to produce a recovered pressure $s_h$, leading to an improved interface velocity via a two-point flux computation. Numerical experiments on homogeneous and heterogeneous $\mathbf{K}$ demonstrate notable gains in the convergence rate of the interface velocity (e.g., from $O(h^{0.95})$ to $O(h^{1.5})$), supporting its use for better velocity approximations and as a basis for recovery-based error estimates in coupled flow–transport problems. The approach provides a practical mechanism to enhance velocity accuracy while maintaining a domain-decomposition framework, with potential impact on a posteriori error analysis and adaptive strategies.

Abstract

The velocity, coupling term in the flow and transport problems, is important in the accurate numerical simulation or in the posteriori error analysis for adaptive mesh refinement. We consider Enhanced Velocity Mixed Finite Element Method for the incompressible Darcy flow. In this paper, our aim to study the improvement of velocity at interface to achieve the better approximation of velocity between subdomains. We propose the reconstruction of velocity at interface by using the post-processed pressure. Numerical results at the interface show improvement on convergence rate.

Figures (6)

• Figure 1: Illustration of a domain $\Omega$ with subdomains $\Omega_i$ and non-matching mesh discretization $\mathcal{T}_h$.
• Figure 2: Degrees of freedom for the Enhanced Velocity space.
• Figure 3: The spatial domain and illustration of Enhanced Velocity values on the interface
• Figure 4: Example of left ($\Omega_L$) and right ($\Omega_R$) domains with ghost layers ghost layers
• Figure 5: The illustration of the velocity improvement at the interface using postprocessing.