Special Session 163: Mathematical Modeling of Multiphysics Coupled Systems—Models, Algorithms, and Scalable Computing

Mixed Finite Element Methods for Boundary Control Problems Constrained by the Biharmonic Equation

Giorgio Bornia Alma Mater Studiorum University of Bologna Italy Co-Author(s):

Abstract: We study boundary optimal control problems governed by the biharmonic equation, motivated by applications in thin plate deformation and structural optimization. The presence of boundary controls introduces analytical and numerical challenges due to the involvement of fractional Sobolev trace spaces. In particular, the control variable is defined on the boundary and naturally belongs to fractional spaces, which complicates both the functional setting and the numerical approximation. Additionally, the fourth-order structure of the biharmonic operator requires higher regularity of the state variable, posing a challenge on classical conforming discretizations. To address these difficulties, we employ a lifting strategy that decomposes the state variable into a homogeneous component and a lifting function encoding the boundary control. This reformulation transforms the original boundary control problem into a distributed one, thereby avoiding the explicit treatment of fractional spaces. The resulting system is discretized using a Hermann--Miyoshi mixed finite element method. This approach reduces the fourth-order equation to a system of lower-order equations and allows the use of standard finite element spaces. We analyze the well-posedness of the continuous optimal control problem and derive a first-order necessary condition. Numerical experiments are presented to demonstrate the effectiveness of the formulation

Unconditionally energy-stable, and fully discrete finite element schemes for the Rosensweig model

Xiaojing Dong Xiangtan University Peoples Rep of China Co-Author(s):    Xiaojing Dong, Huayi Huang,Yunqing Huang

Abstract: Ferrohydrodynamics (FHD) describes the motion of a magnetic fluid, usually called a ferrofluid. A ferrofluid is a stable colloidal fluid consisting of nanoscale ferromagnetic particles suspended in a carrier fluid. A colloidal ferrofluid can keep magnetization and fluidity under the action of an external magnetic field.The constitutive equation we consider, proposed by Rosensweig, models fluid dynamics, spins of ferromagnetic particles, magnetic polarization, and a magnetic induction field. The corresponding model incorporates the Navier-Stokes equations, the angular momentum equation, the magnetization equation, and the magnetostatic equation. In this talk, we propose linear, unconditionally energy-stable, and fully discrete finite element schemes for the model. We obtain the existence and uniqueness of the numerical solutions by the Leray-Schauder fixed point theorem, and prove the unconditional convergence through the Aubin-Lions-Simon lemma. Numerical experiments verify the effectiveness and accuracy of the schemes, and simulate the controllability of the magnetic fluid driven by an applied magnetic field.

Neural enrichment finite element method: A hybrid method for problems with strong oscillations and interface problems

Shihan Guo Otto von Guericke University Magdeburg Germany Co-Author(s):    Thomas Richter

Abstract: We propose a hybrid method, the Neural Enrichment Finite Element Method (NEFEM), designed for problems involving strong oscillations or discontinuities. This method is based on stable generalized FEM (SGFEM) framework, wherein neural networks (NNs) are introduced as enrichment functions for adaptivity, and the Ritz functional is applied for training process. The advancements are twofolds. First, the method constructs local subspaces with superior approximation properties, significantly reducing the required number of degrees of freedom (DoFs). Second, minimal \textit{a priori} knowledge is required to define enrichment functions, as the NNs evolve heuristically during training. Furthermore, for smooth problems, we provide a residual-based error estimator and prove both its reliability and efficiency. For interface problems, a theoretical analysis on the optimal convergence of the stable GFEM is studied, notably without imposing additional regularity assumptions. These analysis results guide the network architecture design and training strategies. The performance and effectiveness of the proposed method are validated through several numerical experiments.

Novel unconditionally stable regularization schemes for the Navier-Stokes equations

Ping Lin University of Dundee Scotland Co-Author(s):

Abstract: In the talk we report novel first- and second-order decoupled schemes for the Navier-Stokes equations based on the penalty method and the sequential regularization method (SRM), respectively. These schemes do not require the boundary condition on the pressure and thus preserve the original velocity boundary conditions. By using the idea of the scalar auxiliary variable (SAV), the nonlinear terms of these schemes are treated explicitly. We also carefully reformulated the Navier-Stokes equations to ensure convergence of the proposed scheme without any restriction on the time step. At each time step it is only necessary to solve elliptic equations with constant coefficients. An unconditional (without time step constraints) global optimal error estimate is shown. Furthermore, to more accurately approximate the incompressibility constraint without introducing extra stiffness into the system, a sequential regularization SAV scheme is developed, and is error estimate is provided as well. Finally, we compare our proposed schemes with the classic linearized projection scheme to demonstrate its accuracy and efficiency, and discuss the application of the idea to phase-field models. This is a joint work with Z Wang.

Fast Operator-Splitting Methods for Nonlinear Elliptic Equations

Hao Liu Hong Kong Baptist University Hong Kong Co-Author(s):    Jingyu Yang, Shingyu Leung, Jianliang Qian

Abstract: Nonlinear elliptic problems arise in many fields, including plasma physics, astrophysics, and optimal transport. In this talk, we present our recent work on a novel operator-splitting/finite element method for solving such problems. We begin by introducing an auxiliary function in a new way for a semilinear elliptic partial differential equation, leading to the development of a convergent operator-splitting/finite element scheme for this equation. The algorithm is then extended to fully nonlinear elliptic equations of the Monge-Ampere type, including the Dirichlet Monge-Ampere equation and Pucci`s equation. This is achieved by reformulating the fully nonlinear equations into forms analogous to the semilinear case, enabling the application of the proposed splitting algorithm. In our implementation, a mixed finite element method is used to approximate both the solution and its Hessian matrix. Numerical experiments show that the proposed method outperforms existing approaches in efficiency and accuracy, and can be readily applied to problems defined on domains with curved boundaries.

A Priori and a Posteriori Error Estimate for Pressure Robust Schemes for Incompressible Flow

Lin Mu University of Georgia USA Co-Author(s):

Abstract: The incompressible fluid model is widely used in various fields in engineering and science and their numerical solutions are of prominent importance in understanding complex, natural, engineered, and societal systems. There has been considerable interest in mathematical modeling and algorithm development. One of the critical challenges is the development of the pressure robust scheme and achieve the desired mass conservation with low cost. Our effort aims at designing the low-cost divergence preserving finite element method and in turn, achieve viscosity independent velocity error estimates. Translating this result to the incompressible fluid equations, our algorithm is robust with varying viscosity permeability values and large pressure gradients. In this talk, we shall present our algorithm development, and then demonstrate the stability and convergence analysis theoretically and numerically. The profiles of benchmark tests indicate that our algorithm outperforms other non-divergence preserving numerical schemes.

Ensemble decoupled algorithms for dual-porosity-Stokes model with random physical parameters

Li Shan Shantou University Peoples Rep of China Co-Author(s):

Abstract: The coupling of dual porosity seepage flow with free flow arises in modeling fractured porous media with large conduits. Those problems are encountered in many applications,like petroleum extraction, hydrology, geothermal systems and so on. However, accurate simulations are usually not feasible due to the fact it is physically impossible to know the exact parameter values in the model of interest. In this talk, we focus on the fast simulation for the dual-porosity-Stokes system with random physical parameters. Based on ensemble idea, we propose three decoupled algorithms and provide their stabilities and optimal error estimations. Besides, the numerical tests verify our theoretical results.

A RBF Meshless Galerkin Method for Elliptic Dirichlet Boundary Control on Curved Domains

Yizhong Sun Hong Kong Baptist University Peoples Rep of China Co-Author(s):    Leevan Ling, Shu Ma, Weifeng Qiu, Yizhong Sun

Abstract: We develop and analyze a RBF meshless Galerkin method for elliptic Dirichlet boundary control problems on smooth curved domains. The method is posed on the exact curved domain and imposes the inhomogeneous Dirichlet boundary condition weakly by a Nitsche-type formulation. This avoids the geometric errors introduced when curved boundaries are approximated by polynomial boundary-fitted meshes. Using Bernstein inequalities for kernel-based trial spaces, we derive kernel-based inverse and trace estimates tailored to this boundary control setting. These estimates allow us to prove the stability and optimal error estimates of the proposed method. Numerical experiments on smooth domains with curved boundaries confirm the theoretical convergence rates. The approach can also provides a basis for extensions to curved interface problems, in particular to multi-domain coupled problems with curved interfaces.

Explicit Splitting Scheme for Fluid-Poroelastic Structure Interaction Problems and its Error Analysis

Yifan Wang Texas Tech University USA Co-Author(s):    Yifan Wang, Jeonghun Lee, Suncica Canic

Abstract: \begin{abstract} We present an \emph{a priori} error analysis for a fully discrete, parallelizable, explicitly coupled splitting scheme for fluid--poroelastic structure interaction problems modeled by the time-dependent Stokes--Biot system. The method decouples the fluid and poroelastic subproblems in a fully explicit manner, enabling the two systems to be solved independently at each time step while consistently enforcing the interface conditions. This structure makes the scheme computationally efficient and well suited for partitioned implementations. \newlines The analysis is carried out within a discrete energy framework. We introduce Ritz-type projections in the fluid and poroelastic subdomains and compare the fully discrete scheme with a time-discrete continuous formulation. This yields reduced error equations in which the leading interpolation terms cancel, leaving only consistency errors associated with temporal discretization and lagged interface data arising from the explicit splitting. Using these reduced equations, we derive a discrete error energy identity and establish unconditional error estimates in a combined energy--dissipation norm through a Gronwall-type argument. The resulting bounds prove first-order accuracy in time and optimal convergence in space, with rates determined by the polynomial degree of the finite element spaces. Numerical experiments based on manufactured solutions confirm the theoretical results, showing first-order temporal convergence for all variables and optimal spatial convergence rates. \end{abstract}

A Locking-free and Loosely Coupled Robin-Robin Scheme for Fluid-Poroelasticity Interaction

Xiaohe Yue Leibniz University Hannover Germany Co-Author(s):    Wenlong He, Thomas Wick, Xiaohe Yue, Jiwei Zhang, Haibiao Zheng

Abstract: We propose a loosely coupled scheme for the incompressible fluid-fully dynamic poroelasticity interaction (FPSI) problem, which is locking-free for extreme model parameters. By introducing two new auxiliary variables, the poroelastic system is reformulated into a four-field formulation, which can be regarded as a coupling of a dynamic Stokes-like system and a diffusion system. The decoupled scheme is constructed based on Robin-Robin type coupling conditions, evaluating the right-hand side of these conditions at the previous time step. The resulting scheme is fully parallel, meaning that the fluid subsystem and poroelasticity subsystem can be solved independently and concurrently at each time step without expensive sub-iterations. Furthermore, we prove that the scheme is unconditionally stable and provide an error estimate showing that the convergence in the $H^1$ norm is optimal. The error analysis demonstrates that our scheme is robust against extreme model parameters, thereby eliminating the locking inherent in the poroelastic system. Various numerical experiments are presented to validate the stability, convergence and locking-free performance of the proposed scheme.

General numerical framework to derive structure preserving reduced order models for thermodynamically consistent reversible-irreversible PDEs

Jia Zhao University of Alabama USA Co-Author(s):

Abstract: In this talk, I will present a general numerical framework to derive structure-preserving reduced-order models for thermodynamically consistent PDEs. Our numerical framework has two primary features: (a) a systematic way to extract reduced-order models for thermodynamically consistent PDE systems while maintaining their inherent thermodynamic principles and (b) a general process to derive accurate, efficient, and structure-preserving numerical algorithms to solve these reduced-order models.