arXiv subjects

Hongxing Rui

Publications and source records attributed to Hongxing Rui.

15 recordsLinked to original sources

arXiv2026

The MAC scheme for linear elasticity in displacement-stress formulation on non-uniform staggered grids

A marker-and-cell finite difference method is developed for solving the two dimensional and three dimensional linear elasticity in the displacement-stress formulation on staggered grids. The method employs a staggered grid arrangement, where the displacement components are approximated on the midpoints of cell edges, the normal stresses are defined at the cell centers, and the shear stresses are defined at the grid points. This structure ensures local conservation properties and avoids spurious oscillations in stress approximation. A rigorous mathematical analysis is presented, establishing the stability of the scheme and proving the second-order L2-error super-convergence for both displacement and stress. The proposed method is locking-free with respect to the Lame constant, making it suitable for both compressible and nearly incompressible elastic materials. Numerical experiments demonstrate the efficiency and robustness of the finite difference scheme, and the computed results show excellent agreement with the theoretical predictions.

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arXiv2025

Coupling of conforming and mixed finite element methods for a model of wave propagation in thermo-poroelasticity in the frequency domain

A dynamic linear thermo-poroelasticity model, containing inertial and relaxation terms with second-order time derivatives, is investigated in this paper. The mathematical and numerical analysis of this model is performed in the frequency domain. The variational formulation is analyzed within the framework of Fredholm's alternative and T-coercivity. Under appropriate assumptions on the coefficients, the well-posedness of the problem is proved. For its discretization, we propose a stabilized coupling of conforming and mixed finite element spaces, which are free of volumetric locking, and both, pressure as well as temperature oscillations. By incorporating projections in certain sesquilinear forms, the well-posedness of the finite element solution can be obtained through a similar reasoning as in the continuous case. Optimal error estimates are derived for all variables. Numerical studies validate the accuracy and robustness of the proposed method.

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arXiv2025

A framework of discontinuous Galerkin neural networks for iteratively approximating residuals

We propose an abstract discontinuous Galerkin neural network (DGNN) framework for analyzing the convergence of least-squares methods based on the residual minimization when feasible solutions are neural networks. Within this framework, we define a quadratic loss functional as in the least square method with $h-$refinement and introduce new discretization sets spanned by element-wise neural network functions. The desired neural network approximate solution is recursively supplemented by solving a sequence of quasi-minimization problems associated with the underlying loss functionals and the adaptively augmented discontinuous neural network sets without the assumption on the boundedness of the neural network parameters. We further propose a discontinuous Galerkin Trefftz neural network discretization (DGTNN) only with a single hidden layer to reduce the computational costs. Moreover, we design a template based on the considered models for initializing nonlinear weights. Numerical experiments confirm that compared to existing PINN algorithms, the proposed DGNN method with one or two hidden layers is able to improve the relative $L^2$ error by at least one order of magnitude at low computational costs.

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arXiv2025

A linear, mass-conserving, multi-time-step compact block-centered finite difference method for incompressible miscible displacement problem in porous media

In this paper, a two-dimensional incompressible miscible displacement model is considered, and a novel decoupled and linearized high-order finite difference scheme is developed, by utilizing the multi-time-step strategy to treat the different time evolutions of concentration and velocity/pressure, and the compact block-centered finite difference approximation for spatial discretization. We show that the scheme is mass-conserving, and has second-order temporal accuracy and fourth-order spatial accuracy for the concentration, the velocity and the pressure simultaneously. The existence and uniqueness of the developed scheme under a rough time-step condition is also proved following the convergence results. Numerical experiments are presented to confirm the theoretical conclusions. Besides, some 'real' simulations are also tested to show good performance of the proposed scheme, in particular, the viscous fingering phenomenon is verified.

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arXiv2025

Robust superconvergence analysis of physics-preserving RMAC scheme for the Stokes and Navier--Stokes equations on non-uniform grids at high Reynolds numbers

The velocity errors of the classical marker and cell (MAC) scheme are dependent on the pressure approximation errors, which is non-pressure-robust and will cause the accuracy of the velocity approximation to deteriorate when the pressure approximation is poor. In this paper, we first propose the reconstructed MAC scheme (RMAC) based on the finite volume method to obtain the pressure-robustness for the time-dependent Stokes equations and then construct the $\mu$-robust and physics-preserving RMAC scheme on non-uniform grids for the Navier--Stokes equations, where $\mu$-robustness means that the velocity errors do not blow up for small viscosity $\mu$ when the true velocity is sufficiently smooth. Compared with the original MAC scheme, which was analyzed in [SIAM J. Numer. Anal. 55 (2017): 1135-1158], the RMAC scheme is different only on the right-hand side for Stokes equations. It can also be proved that the constructed scheme satisfies the local mass conservation law, the discrete unconditional energy dissipation law, the momentum conservation, and the angular momentum conservation for the Stokes and Navier--Stokes equations. Furthermore, by constructing the new auxiliary function depending on the velocity and using the high-order consistency analysis, we can obtain the pressure-robust and $\mu$-robust error estimates for the velocity and derive the second-order superconvergence for the velocity and pressure in the discrete $l^{\infty}(l^2)$ norm on non-uniform grids and the discrete $l^{\infty}(l^{\infty})$ norm on uniform grids. Finally, numerical experiments using the constructed schemes are demonstrated to show the robustness for our constructed schemes.

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arXiv2024

Analysis of a $\boldsymbol{P}_1\oplus \boldsymbol{RT}_0$ finite element method for linear elasticity with Dirichlet and mixed boundary conditions

In this paper, we investigate a low-order robust numerical method for the linear elasticity problem. The method is based on a Bernardi--Raugel-like $\boldsymbol{H}(\mathrm{div})$-conforming method proposed first for the Stokes flows in [Li and Rui, IMA J. Numer. Anal. {42} (2022) 3711--3734].Therein the lowest-order $\boldsymbol{H}(\mathrm{div})$-conforming Raviart--Thomas space ($\boldsymbol{RT}_0$) was added to the classical conforming $\boldsymbol{P}_1\times P_0$ pair to meet the inf-sup condition, while preserving the divergence constraint and some important features of conforming methods. Due to the inf-sup stability of {the} $\boldsymbol{P}_1\oplus \boldsymbol{RT}_0\times P_0$ pair, a locking-free elasticity discretization {with respect to} {the Lam\'{e} constant $\lambda$} can be naturally obtained. Moreover, our scheme is gradient-robust for the pure and homogeneous displacement boundary problem, that is, the discrete $\boldsymbol{H}^1$-norm of the displacement is $\mathcal{O}(\lambda^{-1})$ when the external body force is a gradient field. We also consider the mixed displacement and stress boundary problem, whose $\boldsymbol{P}_1\oplus \boldsymbol{RT}_0$ discretization should be carefully designed due to a consistency error arising from the $\boldsymbol{RT}_0$ part. We propose both symmetric and nonsymmetric schemes to approximate the mixed boundary case. The optimal error estimates are derived for the energy norm and/or $\boldsymbol{L}^2$-norm. Numerical experiments demonstrate the accuracy and robustness of our schemes.

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arXiv2023

On efficient linear and fully decoupled finite difference method for wormhole propagation with heat transmission process on staggered grids

In this paper, we construct an efficient linear and fully decoupled finite difference scheme for wormhole propagation with heat transmission process on staggered grids, which only requires solving a sequence of linear elliptic equations at each time step. We first derive the positivity preserving properties for the discrete porosity and its difference quotient in time, and then obtain optimal error estimates for the velocity, pressure, concentration, porosity and temperature in different norms rigorously and carefully by establishing several auxiliary lemmas for the highly coupled nonlinear system. Numerical experiments in two- and three-dimensional cases are provided to verify our theoretical results and illustrate the capabilities of the constructed method.

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arXiv2022

Inf-sup stabilized Scott--Vogelius pairs on general simplicial grids by Raviart--Thomas enrichment

This paper considers the discretization of the Stokes equations with Scott--Vogelius pairs of finite element spaces on arbitrary shape-regular simplicial grids. A novel way of stabilizing these pairs with respect to the discrete inf-sup condition is proposed and analyzed. The key idea consists in enriching the continuous polynomials of order $k$ of the Scott--Vogelius velocity space with appropriately chosen and explicitly given Raviart--Thomas bubbles. This approach is inspired by [Li/Rui, IMA J. Numer. Anal, 2021], where the case $k=1$ was studied. The proposed method is pressure-robust, with optimally converging $\boldsymbol{H}^1$-conforming velocity and a small $\boldsymbol{H}(\mathrm{div})$-conforming correction rendering the full velocity divergence-free. For $k\ge d$, with $d$ being the dimension, the method is parameter-free. Furthermore, it is shown that the additional degrees of freedom for the Raviart--Thomas enrichment and also all non-constant pressure degrees of freedom can be condensated, effectively leading to a pressure-robust, inf-sup stable, optimally convergent $\boldsymbol{P}_k \times P_0$ scheme. Aspects of the implementation are discussed and numerical studies confirm the analytic results.

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arXiv2021

Longer time simulation of the unsteady Navier-Stokes equations based on a modified convective formulation

For the discretization of the convective term in the Navier-Stokes equations (NSEs), the commonly used convective formulation (CONV) does not preserve the energy if the divergence constraint is only weakly enforced. In this paper, we apply the skew-symmetrization technique in [B. Cockburn, G. Kanschat and D. Schötzau, Math. Comp., 74 (2005), pp. 1067-1095] to conforming finite element methods, which restores energy conservation for CONV. The crucial idea is to replace the discrete advective velocity with its a $H(\operatorname{div})$-conforming divergence-free approximation in CONV. We prove that the modified convective formulation also conserves linear momentum, helicity, 2D enstrophy and total vorticity under some appropriate senses. Its a Picard-type linearization form also conserves them. Under the assumption $\boldsymbol{u}\in L^{2}(0,T;\boldsymbol{W}^{1,\infty}(Ω)),$ it can be shown that the Gronwall constant does not explicitly depend on the Reynolds number in the error estimates. The long time numerical simulations show that the linearized and modified convective formulation has a similar performance with the EMAC formulation and outperforms the usual skew-symmetric formulation (SKEW).

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arXiv2021

An EMA-conserving, pressure-robust and Re-semi-robust reconstruction method for the unsteady incompressible Navier-Stokes equations

Proper EMA-balance (E: kinetic energy; M: momentum; A: angular momentum), pressure-robustness and $Re$-semi-robustness ($Re$: Reynolds number) are three important properties of Navier-Stokes simulations with exactly divergence-free elements. This EMA-balance makes a method conserve kinetic energy, linear momentum and angular momentum under some suitable senses; pressure-robustness means that the velocity errors are independent of the continuous pressure; $Re$-semi-robustness means that the constants appearing in the error bounds of kinetic and dissipation energies do not explicitly depend on inverse powers of the viscosity. In this paper, based on the pressure-robust reconstruction methods in [{A. Linke and C. Merdon, {\it Comput. Methods Appl. Mech. Engrg.} 311 (2016), 304-326}], we propose a novel reconstruction method for a class of non-divergence-free simplicial elements which admits almost all the above properties. The only exception is the energy balance, where kinetic energy should be replaced by a properly redefined discrete energy. Some numerical comparisons with exactly divergence-free methods, pressure-robust reconstructions and the EMAC scheme are provided to confirm our theoretical results.

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arXiv2021

A low order divergence-free H(div)-conforming finite element method for Stokes flows

In this paper, we propose a ${ P_{1}^{c}}\oplus {RT0}-P0$ discretization of the Stokes equations on general simplicial meshes in two/three dimensions (2D/3D), which yields an exactly divergence-free and pressure-independent velocity approximation with optimal order. Our method has the following features. Firstly, the global number of the degrees of freedom of our method is the same as the low order Bernardi and Raugel ($B$-$R$) finite element method (Bernardi and Raugel, 1985), while the number of {the non-zero entries} of the former is about half of the latter in the velocity-velocity region of the coefficient matrix. Secondly, the ${ P_{1}^{c}}$ component of the velocity, the $RT0$ component of the velocity and the pressure seem to solve a popular ${ P_{1}^{c}}-{RT0}-P0$ discretization of a poroelastic-type system formally. Finally, our method can be easily transformed into a pressure-robust and stabilized ${ P_{1}^{c}}-P0$ discretization for the Stokes problem via the static condensation of the $RT0$ component, which has a much smaller number of global degrees of freedom. Numerical experiments illustrating the robustness of our method are also provided.

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arXiv2020

A Stabilized Hybrid Mixed Finite Element Method for Poroelasticity

In this work, we consider a hybrid mixed finite element method for Biot's model. The hybrid P1-RT0-P0 discretization of the displacement-pressure-Darcy's velocity system of Biot's model presented in \cite{C. Niu} is not uniformly stable with respect to the physical parameters, resulting in some issues in numerical simulations. To alleviate such problems, following \cite{V. Girault}, we stabilize the hybrid scheme with face bubble functions and show the well-posedness with respect to physical and discretization parameters, which provide optimal error estimates of the stabilized method. We introduce a perturbation of the bilinear form of the displacement which allows for the elimination of the bubble functions. Together with eliminating Darcy's velocity by hybridization, we obtain an eliminated system whose size is the same as the classical P1-RT0-P0 discretization. Based on the well-posedness of the eliminated system, we design block preconditioners that are parameter-robust. Numerical experiments are presented to confirm the theoretical results of the stabilized scheme as well as the block preconditioners.

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arXiv2018

Energy Stability and Convergence of SAV Block-centered Finite Difference Method for Gradient Flows

We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.

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arXiv2016

Multigrid Methods for A Mixed Finite Element Method of The Darcy-Forchheimer Model

An efficient nonlinear multigrid method for a mixed finite element method of the Darcy-Forchheimer model is constructed in this paper. A Peaceman-Rachford type iteration is used as a smoother to decouple the nonlinearity from the divergence constraint. The nonlinear equation can be solved element-wise with a closed formulae. The linear saddle point system for the constraint is reduced into a symmetric positive definite system of Poisson type. Furthermore an empirical choice of the parameter used in the splitting is proposed and the resulting multigrid method is robust to the so-called Forchheimer number which controls the strength of the nonlinearity. By comparing the number of iterations and CPU time of different solvers in several numerical experiments, our multigrid method is shown to convergent with a rate independent of the mesh size and the Forchheimer number and with a nearly linear computational cost.

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arXiv2014

Convergence of a decoupled mixed FEM for miscible displacement in interfacial porous media

In this paper, we study the stability and convergence of a decoupled and linearized mixed finite element method (FEM) for incompressible miscible displacement in a porous media whose permeability and porosity are discontinuous across some interfaces. We show that the proposed scheme has optimal-order convergence rate unconditionally, without restriction on the grid ratio (between the time-step size and spatial mesh size). Previous works all required certain restrictions on the grid ratio except for the problem with globally smooth permeability and porosity. Our idea is to introduce an intermediate system of elliptic interface problems, whose solution is uniformly regular in each subdomain separated by the interfaces and its finite element solution coincides with the fully discrete solution of the original problem. In order to prove the boundedness of the fully discrete solution, we study the finite element discretization of the intermediate system of elliptic interface problems.

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