arXiv subjects

Huangxin Chen

Publications and source records attributed to Huangxin Chen.

At least 19 recordsLinked to original sources

arXiv2026

A unified analysis of maximum-norm estimates for a class of HDG methods for parabolic problem in polyhedral domains

This paper studies a general class of semi-discrete hybridizable discontinuous Galerkin (HDG) methods, including mixed methods, for parabolic problems in nonconvex polygonal and polyhedral domains. By developing local energy error estimates together with energy estimates for a regularized Green's function, we establish a unified framework to prove the maximum-norm stability of both the semigroup defined by the semi-discrete scheme and the corresponding discrete solutions. The stability analysis shows that the main challenges stem from the treatment of numerical flux variables and the inherent asymmetry of the discrete scheme. These challenges are intrinsic to numerical approaches formulated within the mixed framework for parabolic equations. The asymmetry prevents the direct application of the double kick-back argument, while the presence of flux variables requires special techniques to control their values at the initial time. We emphasize that the analytical tools developed here to address these challenges are sufficiently general to be adapted for maximum-norm stability investigations of a broader class of numerical schemes arising from mixed formulations of parabolic equations. Furthermore, by the stability results and their proofs, we derive the maximal regularity of the semi-discrete solution in $L^{\infty}((0,T);L^p(Ω))$-norm and reduce the maximum-norm error estimates to those of the corresponding elliptic equations and the $L^2$-orthogonal projection. Since the derivation of the local energy error estimates does not rely on the lifting operator, which is only available for simplicial meshes, our results (excluding mixed methods) remain valid for polygonal/polyhedral meshes.

math.NA↗

arXiv2026

An Energy-Stable, Bound-Preserving and Locally Conservative Numerical Framework for Multicomponent Gas Flow in Poroelastic Media

In this paper, we propose a robust and efficient numerical framework for simulating multicomponent gas flow in poroelastic media, with a focus on preserving fundamental thermodynamic principles and ensuring computational reliability. The model captures the complex nonlinear coupling between multicomponent transport and solid deformation, while addressing critical numerical challenges such as mass conservation, energy stability, and molar density boundedness. To achieve this, we develop a stabilized discretization approach that guarantees the preservation of the original energy dissipation law and ensures the boundedness of each gas component's molar density. Furthermore, the proposed method incorporates an adaptive time-stepping strategy that dynamically adjusts the time step size based on the system's dynamics, significantly enhancing computational efficiency without compromising stability or accuracy. For spatial discretization, a mixed finite element method combined with an upwind scheme is employed for the flow and transport equations to ensure local mass conservation, while a discontinuous Galerkin (DG) method is utilized for discretizing the momentum equation of poroelasticity to effectively overcome numerical locking phenomena. Numerical experiments are presented to demonstrate the performance, robustness, and applicability of the method in simulating multicomponent gas flow under various scenarios.

math.NA↗

arXiv2026

Bound-Preserving Adaptive Time-Stepping Method with Energy Stability for Simulating Compressible Gas Flow in Poroelastic Media

In this paper, we present an efficient numerical method to address a thermodynamically consistent gas flow model in porous media involving compressible gas and deformable rock. The accurate modeling of gas flow in porous media often poses significant challenges due to their inherent nonlinearity, the coupling between gas and rock dynamics, and the need to preserve physical principles such as mass conservation, energy dissipation and molar density boundedness. The system is further complicated by the need to balance computational efficiency with the accuracy and stability of the numerical scheme. To tackle these challenges, we adopt a stabilization approach that is able to preserve the original energy dissipation while achieving linear energy-stable numerical schemes. We also prove the convergence of the adopted linear iterative method. At each time step, the stabilization parameter is adaptively updated using a simple and explicit formula to ensure compliance with the original energy dissipation law. The proposed method uses adaptive time stepping to improve computational efficiency while maintaining solution accuracy and boundedness. The adaptive time step size is calculated explicitly at each iteration, ensuring stability and allowing for efficient handling of highly dynamic scenarios. A mixed finite element method combined with an upwind scheme is employed as spatial discretization to ensure mass conservation and stability. Finally, we conduct a series of numerical experiments to validate the performance and robustness of the proposed numerical method.

math.NA↗

arXiv2026

The Existence, uniqueness, and regularity of weak solutions for a thermodynamically consistent two-phase flow model in porous media

Thermodynamically consistent models for two-phase flow in porous media have attracted significant attention in recent years. In this paper, we prove the existence, uniqueness and regularity of the weak solution to such a recent model proposed in [25,35]. To this end, firstly, we introduce a fully implicit time semi-discrete approximation and a fully discrete approximation for an appropriate weak formulation of the thermodynamically consistent model. Next, by using the zeros of a vector field theorem, we prove the existence of the weak solution for the fully discrete approximation. Then the existence of weak solutions for the fully implicit time semi-discrete approximation and the weak formulation of the model are derived by the weak convergence technique and the energy stability estimate. Subsequently, by the Gr{\" o}nwall inequality, we prove the uniqueness result under the smoothness assumption on the chemical potential. Finally, combined with the regularity theory of elliptic partial differential equations (PDE), the regularity of the weak solution for the model with complete Neumann boundary conditions is established.

math.AP↗

arXiv2025

An Optimally Convergent parallel splitting Algorithm for the Multiple-Network Poroelasticity Model

This paper presents a novel parallel splitting algorithm for solving quasi-static multiple-network poroelasticity (MPET) equations. By introducing a total pressure variable, the MPET system can be reformulated into a coupled Stokes-parabolic system. To efficiently solve this system, we propose a parallel splitting approach. In the first time step, a monolithic solver is used to solve all variables simultaneously. For subsequent time steps, the system is split into a Stokes subproblem and a parabolic subproblem. These subproblems are then solved in parallel using a stabilization technique. This parallel splitting approach differs from sequential or iterative decoupling, significantly reducing computational time. The algorithm is proven to be unconditionally stable, optimally convergent, and robust across various parameter settings. These theoretical results are confirmed by numerical experiments. We also apply this parallel algorithm to simulate fluid-tissue interactions within the physiological environment of the human brain.

math.NA↗

arXiv2025

A robust and stable phase field method for structural topology optimization

This paper presents a novel phase-field-based methodology for solving minimum compliance problems in topology optimization under fixed external loads and body forces. The proposed framework characterizes the optimal structure through an order parameter function, analogous to phase-field models in materials science, where the design domain and its boundary are intrinsically represented by the order parameter function. The topology optimization problem is reformulated as a constrained minimization problem with respect to this order parameter, requiring simultaneous satisfaction of three critical properties: bound preservation, volume conservation, and monotonic objective functional decay throughout the optimization process. The principal mathematical challenge arises from handling domain-dependent body forces, which necessitates the development of a constrained optimization framework. To address this, we develop an operator-splitting algorithm incorporating Lagrange multipliers, enhanced by a novel limiter mechanism. This hybrid approach guarantees strict bound preservation, exact volume conservation, and correct objective functional decaying rate. Numerical implementation demonstrates the scheme's robustness through comprehensive 2D and 3D benchmarks.

math.OC↗

arXiv2024

The Onsager principle and structure preserving numerical schemes

We present a natural framework for constructing energy-stable time discretization schemes. By leveraging the Onsager principle, we demonstrate its efficacy in formulating partial differential equation models for diverse gradient flow systems. Furthermore, this principle provides a robust basis for developing numerical schemes that uphold crucial physical properties. Within this framework, several widely used schemes emerge naturally, showing its versatility and applicability.

math.NA↗

arXiv2024

A locally mass-conservative enriched Petrov-Galerkin method without penalty for the Darcy flow in porous media

In this work we present an enriched Petrov-Galerkin (EPG) method for the simulation of the Darcy flow in porous media. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions, and it does not require any penalty term in the weak formulation. Moreover, we propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method based on this framework, which enables the process of solving pressure to be decoupled into two steps. The first step is to solve the pressure by the standard CG method, and the second step is a post-processing correction of the first step. Compared with the CG method, the proposed EPG method is locally mass-conservative, while keeping fewer degrees of freedom than the discontinuous Galerkin (DG) method. In addition, this method is more concise in the error analysis than the enriched Galerkin (EG) method. The coupled flow and transport in porous media is considered to illustrate the advantages of locally mass-conservative properties of the EPG method. We establish the optimal convergence of numerical solutions and present several numerical examples to illustrate the performance of the proposed method.

math.NA↗

arXiv2024

A prediction-correction based iterative convolution-thresholding method for topology optimization of heat transfer problems

In this paper, we propose an iterative convolution-thresholding method (ICTM) based on prediction-correction for solving the topology optimization problem in steady-state heat transfer equations. The problem is formulated as a constrained minimization problem of the complementary energy, incorporating a perimeter/surface-area regularization term, while satisfying a steady-state heat transfer equation. The decision variables of the optimization problem represent the domains of different materials and are represented by indicator functions. The perimeter/surface-area term of the domain is approximated using Gaussian kernel convolution with indicator functions. In each iteration, the indicator function is updated using a prediction-correction approach. The prediction step is based on the variation of the objective functional by imposing the constraints, while the correction step ensures the monotonically decreasing behavior of the objective functional. Numerical results demonstrate the efficiency and robustness of our proposed method, particularly when compared to classical approaches based on the ICTM.

math.NA↗

arXiv2023

An Unconditionally Energy-Stable and Orthonormality-Preserving Iterative Scheme for the Kohn-Sham Gradient Flow Based Model

We propose an unconditionally energy-stable, orthonormality-preserving, component-wise splitting iterative scheme for the Kohn-Sham gradient flow based model in the electronic structure calculation. We first study the scheme discretized in time but still continuous in space. The component-wise splitting iterative scheme changes one wave function at a time, similar to the Gauss-Seidel iteration for solving a linear equation system. Rigorous mathematical derivations are presented to show our proposed scheme indeed satisfies the desired properties. We then study the fully-discretized scheme, where the space is further approximated by a conforming finite element subspace. For the fully-discretized scheme, not only the preservation of orthogonality and normalization (together we called orthonormalization) can be quickly shown using the same idea as for the semi-discretized scheme, but also the highlight property of the scheme, i.e., the unconditional energy stability can be rigorously proven. The scheme allows us to use large time step sizes and deal with small systems involving only a single wave function during each iteration step. Several numerical experiments are performed to verify the theoretical analysis, where the number of iterations is indeed greatly reduced as compared to similar examples solved by the Kohn-Sham gradient flow based model in the literature.

math.NA↗

arXiv2023

Benchmark modeling and 3D applications of solidification and macro-segregation based on an operator-splitting and fully decoupled scheme with term-wise matrix assembly

The solidification and macro-segregation problem involving unsteady multi-physics and multi-phase fields is typically a complex process with mass, momentum, heat, and species transfers among solid, mushy, and liquid phase regions. The quantitative prediction of phase change, chemical heterogeneities, and multi-phase and multi-component flows plays critical roles in many natural scenarios and industrial applications that involve many disciplines, like material, energy, and even planet science. In view of this, some scholars and research institutions have called for more contributors to join the benchmark analysis of solidification and segregation problems. Our work proposes an operator-splitting and matrix-based method to avoid non-linear systems. Also, the combination of vectorization and forward equation-based matrix assembly techniques enhances the implementability of extensions of 3D applications. Lastly, the novel scheme is well validated through a bunch of 2D and 3D benchmark cases. The numerical results also illustrate that this method can ensure accurate prediction and adequately capture the physical details of phenomena caused by the solutally and thermally driven flow, which include channel segregation, the formation of freckles, edge effect, aspect ratio effect, and 3D effect.

physics.flu-dyn↗

arXiv2022

A $C^{0}$ interior penalty method for $m$th-Laplace equation

In this paper, we propose a $C^{0}$ interior penalty method for $m$th-Laplace equation on bounded Lipschitz polyhedral domain in $\mathbb{R}^{d}$, where $m$ and $d$ can be any positive integers. The standard $H^{1}$-conforming piecewise $r$-th order polynomial space is used to approximate the exact solution $u$, where $r$ can be any integer greater than or equal to $m$. Unlike the interior penalty method in [T.~Gudi and M.~Neilan, {\em An interior penalty method for a sixth-order elliptic equation}, IMA J. Numer. Anal., \textbf{31(4)} (2011), pp. 1734--1753], we avoid computing $D^{m}$ of numerical solution on each element and high order normal derivatives of numerical solution along mesh interfaces. Therefore our method can be easily implemented. After proving discrete $H^{m}$-norm bounded by the natural energy semi-norm associated with our method, we manage to obtain stability and optimal convergence with respect to discrete $H^{m}$-norm. Numerical experiments validate our theoretical estimate.

math.NA↗

arXiv2020

A mixed finite element scheme for biharmonic equation with variable coefficient and von Kármán equations

In this paper, a new mixed finite element scheme using element-wise stabilization is introduced for the biharmonic equation with variable coefficient on Lipschitz polyhedral domains. The proposed scheme doesn't involve any integration along mesh interfaces. The gradient of the solution is approximated by $H({\rm div})$-conforming $BDM_{k+1}$ element or vector valued Lagrange element with order $k+1$, while the solution is approximated by Lagrange element with order $k+2$ for any $k\geq 0$.This scheme can be easily implemented and produces positive definite linear system. We provide a new discrete $H^{2}$-norm stability, which is useful not only in analysis of this scheme but also in $C^{0}$ interior penalty methods and DG methods. Optimal convergences in both discrete $H^{2}$-norm and $L^{2}$-norm are derived. This scheme with its analysis is further generalized to the von Kármán equations. Finally, numerical results verifying the theoretical estimates of the proposed algorithms are also presented.

math.NA↗

arXiv2020

A fully discrete energy stable scheme for a phase-field moving contact line model with variable densities and viscosities

In this work, we propose a fully discrete energy stable scheme for the phase-field moving contact line model with variable densities and viscosities. The mathematical model consists of a Cahn-Hilliard equation, a Navier-Stokes equation and the generalized Navier boundary condition for the moving contact line. A scalar auxiliary variable is adopted to transform the governing system into an equivalent form, allowing the double well potential to be treated semi-explicitly. A stabilization term is added to balance the explicit nonlinear term originating from the surface energy at fluid-solid interface. A pressure stabilization method is used to decouple the computation of velocity and pressure. Some subtle implicit-explicit treatments are adopted to deal with convention and stress terms. We establish a rigorous proof of energy stability for the proposed time-marching scheme. Then a finite difference method on staggered grids is used to spatially discretize the constructed time-marching scheme. We further prove that the fully discrete scheme also satisfies the discrete energy dissipation law. Numerical results demonstrate accuracy and energy stability of the proposed scheme. Using our numerical scheme, we analyze the contact line dynamics through a shear flow driven droplet sliding case. Three-dimensional droplet spreading is also investigated on a chemically patterned surface. Our numerical simulation accurately predicts the expected energy evolutions and it successfully reproduces expected phenomena that an oil droplet contracts inwards on a hydrophobic zone and spreads outwards quickly on a hydrophilic zone.

physics.comp-ph↗

arXiv2019

A new physics-preserving IMPES scheme for incompressible and immiscible two-phase flow in heterogeneous porous media

In this work we consider a new efficient IMplicit Pressure Explicit Saturation (IMPES) scheme for the simulation of incompressible and immiscible two-phase flow in heterogeneous porous media with capillary pressure. Compared with the conventional IMPES schemes, the new IMPES scheme is inherently physics-preserving, namely, the new algorithm is locally mass conservative for both phases and it also enjoys another appealing feature that the total velocity is continuous in the normal direction. Moreover, the new scheme is unbiased with regard to the two phases and the saturations of both phases are bounds-preserving if the time step size is smaller than a certain value. The key ideas in the new scheme include that the Darcy flows for both phases are rewritten in the formulation based on the total velocity and an auxiliary velocity referring to as the capillary potential gradient, and the total discretized conservation equation is obtained by the summation of the discretized conservation equation for each phase. The upwind strategy is applied to update the saturations explicitly, and the upwind mixed finite element methods are used to solve the pressure-velocity systems which can be decoupled. We also present some interesting examples to demonstrate the efficiency and robustness of the new algorithm.

math.NA↗

arXiv2019

Extended HDG methods for second order elliptic interface problems

In this paper, we propose two arbitrary order eXtended hybridizable Discontinuous Galerkin (X-HDG) methods for second order elliptic interface problems in two and three dimensions. The first X-HDG method applies to any piecewise $C^2$ smooth interface. It uses piecewise polynomials of degrees $k$ $(k>= 1)$ and $k-1$ respectively for the potential and flux approximations in the interior of elements inside the subdomains, and piecewise polynomials of degree $ k$ for the numerical traces of potential on the inter-element boundaries inside the subdomains. Double value numerical traces on the parts of interface inside elements are adopted to deal with the jump condition. The second X-HDG method is a modified version of the first one and applies to any fold line/plane interface, which uses piecewise polynomials of degree $ k-1$ for the numerical traces of potential. The X-HDG methods are of the local elimination property, then lead to reduced systems which only involve the unknowns of numerical traces of potential on the inter-element boundaries and the interface. Optimal error estimates are derived for the flux approximation in $L^2$ norm and for the potential approximation in piecewise $H^1$ seminorm without requiring "sufficiently large" stabilization parameters in the schemes. In addition, error estimation for the potential approximation in $L^2$ norm is performed using dual arguments. Finally, we provide several numerical examples to verify the theoretical results.

math.NA↗

arXiv2018

An efficient threshold dynamics method for topology optimization for fluids

We propose an efficient threshold dynamics method for topology optimization for fluids modeled with the Stokes equation. The proposed algorithm is based on minimization of an objective energy function that consists of the dissipation power in the fluid and the perimeter approximated by nonlocal energy, subject to a fluid volume constraint and the incompressibility condition. We show that the minimization problem can be solved with an iterative scheme in which the Stokes equation is approximated by a Brinkman equation. The indicator functions of the fluid-solid regions are then updated according to simple convolutions followed by a thresholding step. We demonstrate mathematically that the iterative algorithm has the total energy decaying property. The proposed algorithm is simple and easy to implement. A simple adaptive time strategy is also used to accelerate the convergence of the iteration. Extensive numerical experiments in both two and three dimensions show that the proposed iteration algorithm converges in much fewer iterations and is more efficient than many existing methods. In addition, the numerical results show that the algorithm is very robust and insensitive to the initial guess and the parameters in the model.

math.OC↗

arXiv2016

A superconvergent HDG method for the Maxwell equations

We present and analyze a new hybridizable discontinuous Galerkin (HDG) method for the steady state Maxwell equations. In order to make the problem well-posed, a condition of divergence is imposed on the electric field. Then a Lagrange multiplier $p$ is introduced, and the problem becomes the solution of a mixed curl-curl formulation of the Maxwell's problem. We use polynomials of degree $k+1$, $k$, $k$ to approximate $\bfu,\nabla \times \bfu$ and $p$ respectively. In contrast, we only use a non-trivial subspace of polynomials of degree $k+1$ to approximate the numerical tangential trace of the electric field and polynomials of degree $k+1$ to approximate the numerical trace of the Lagrange multiplier on the faces. On the simplicial meshes, a special choice of the stabilization parameters is applied, and the HDG system is shown to be well-posed. Moreover, we show that the convergence rates for $\boldsymbol{u}$ and $\nabla \times \boldsymbol{u}$ are independent of the Lagrange multiplier $p$. If we assume the dual operator of the Maxwell equation on the domain has adequate regularity, we show that the convergence rate for $\boldsymbol{u}$ is $O(h^{k+2})$. From the point of view of degrees of freedom of the globally coupled unknown: numerical trace, this HDG method achieves superconvergence for the electric field without postprocessing. Finally, we show that on general polyhedral elements, by a particular choice of the stabilization parameters again, the HDG system is also well-posed and the superconvergence of the HDG method is derived.

math.NA↗