Shihua Gong

Luyu Cen, Shihua Gong, Euan A. Spence et al.

This paper analyzes the convergence of parallel overlapping domain-decomposition methods with impedance boundary conditions for the time-harmonic Maxwell equations in heterogeneous media. We prove that the parallel iterative method is well-posed in an appropriate function space, and characterize the error propagation operator through impedance-to-impedance maps that describe interactions between neighboring subdomains. For strip domain decompositions, we derive explicit convergence estimates in terms of the norms of the impedance-to-impedance maps. At the discrete level, we develop the finite-element counterpart of these results based on Nédélec-element discretisations. Under the assumption that the discrete impedance-to-impedance maps approximate their continuous counterparts as the mesh is refined, we show that the discrete method inherits the convergence behavior of the continuous method. We illustrate this theory with numerical experiments for strip domain decompositions, and also present numerical experiments for checkerboard domain decompositions that go beyond our theory.

Shihua Gong, Shuonan Wu, Jinchao Xu

In this paper, we present a family of new mixed finite element methods for linear elasticity for both spatial dimensions $n=2,3$, which yields a conforming and strongly symmetric approximation for stress. Applying $\mathcal{P}_{k+1}-\mathcal{P}_k$ as the local approximation for the stress and displacement, the mixed methods achieve the optimal order of convergence for both the stress and displacement when $k \ge n$. For the lower order case $(n-2\le k<n)$, the stability and convergence still hold on some special grids. The proposed mixed methods are efficiently implemented by hybridization, which imposes the inter-element normal continuity of the stress by a Lagrange multiplier. Then, we develop and analyze multilevel solvers for the Schur complement of the hybridized system in the two dimensional case. Provided that no nearly singular vertex on the grids, the proposed solvers are proved to be uniformly convergent with respect to both the grid size and Poisson's ratio. Numerical experiments are provided to validate our theoretical results.

Lingyue Shen, Qi Xin, Yan Chen et al.

This paper presents a quasi-monolithic localized high-order arbitrary Lagrangian-Eulerian (qMLH-ALE) finite element method for multi-scale fluid-structure interaction (FSI) in microfluidic systems. The fluid momentum, the incompressible Neo-Hookean constitutive law, and the left Cauchy-Green tensor $\mathcal{B}$ are assembled into a single implicit system, while the harmonic mesh extension is updated explicitly in a staggered manner. Isoparametric $\mathcal{P}_2$ elements provide third-order geometric approximation of curved fluid-solid interfaces, and a second-order implicit-explicit partitioned Runge-Kutta scheme delivers second-order temporal accuracy without the dissipation of backward Euler. A localized updating strategy confines the moving mesh and the deformation history to a body-fitted sub-domain coupled with a precomputed steady background flow, bridging the scale disparity between local FSI dynamics and the macroscopic microchannel geometry. The Turek-Hron FSI3 benchmark, performed at unit fluid-solid density ratio, reproduces the reference beam-tip amplitude and frequency within $3\%$, confirming stability under the strong added-mass coupling that destabilizes conventional partitioned schemes. Three-dimensional particle-focusing simulations in spiral microchannels further illustrate the framework on long-range multi-scale problems.

Qi Xin, Shihua Gong, Jinchao Xu

The predictive simulation of fluid dynamics in densely packed microfluidic devices, such as Deterministic Lateral Displacement (DLD) arrays, is severely bottlenecked by the stagnation of standard iterative solvers. In this paper, we reveal that this failure is not an algorithmic artifact, but fundamentally rooted in the pre-asymptotic degradation of the pressure-velocity coupling stability. By rigorously analyzing periodic pillar geometries in this generalized lattice framework, we prove that the continuous Ladyzhenskaya-Babuška-Brezzi (LBB) condition, also called the inf-sup constant, deteriorates exactly as $m^{-1}$ up to a positive multiplicative constant, where $m$ is the pillar density (the number of pillars per unit length). This causes a severe a priori error amplification and extreme ill-conditioning in Schur complement of the saddle point system. To overcome this theoretical limit, we propose a parameter-free, adaptively scaled Augmented Lagrangian (AL) stabilization strategy. Extensive numerical experiments on both standard square and highly asymmetric DLD arrays validate our theoretical bounds and demonstrate the robustness of the proposed AL method.

Shuonan Wu, Shihua Gong, Jinchao Xu

We propose two classes of mixed finite elements for linear elasticity of any order, with interior penalty for nonconforming symmetric stress approximation. One key point of our method is to introduce some appropriate nonconforming face-bubble spaces based on the local decomposition of discrete symmetric tensors, with which the stability can be easily established. We prove the optimal error estimate for both displacement and stress by adding an interior penalty term. The elements are easy to be implemented thanks to the explicit formulations of its basis functions. Moreover, the methods can be applied to arbitrary simplicial grids for any spatial dimension in a unified fashion. Numerical tests for both 2D and 3D are provided to validate our theoretical results.