召集人：Susanne C. Brenner（Louisiana State University，教授）、郑伟英（中国科学院数学与系统科学研究院，研究员）、龚伟（中国科学院数学与系统科学研究院，研究员）

时间：2025.06.01—2025.06.07

Program

June 1-7, 2025    Tianyuan Mathematics Research Center, Kunming, Yunnan Province

Titles and Abstracts

 Convergence of a damped Newton's method for discrete Monge-Ampere functions with a prescribed asymptotic cone

Gerard Awanou

University of Illinois, Chicago, USA

Abstracts: We discuss uniqueness for finite difference discretizations with linear complexity and provably convergent to weak solutions of the second boundary value problem for the Monge-Ampere equation.  The boundary condition is enforced through the use of the notion of asymptotic cone. The differential operator is discretized based on a discrete analogue of the subdifferential. We first prove the convergence of a damped Newton's method for the nonlinear system resulting from the discretization leading to the existence of a solution. For now, we have made an assumption, supported by numerical experiments, that for a certain class of stencils, the direct graph of the Jacobian matrix is connected. With related arguments we prove that such a solution is necessarily unique. Convergence of the discretization as well as numerical experiments are given. 

Three classical fourth-order semilinear problems

Carsten Carstensen

Humboldt University of Berlin, Germany

Abstracts：The general and short title might be better specified and then stands for the mathematical foundation of piecewise quadratic non-conforming and fifth-order conforming finite element methods for nonlinear smooth low-order perturbations of the biharmonic equation. There is in fact a competition of low-budget nonconforming methodologies like the popular Morley, WOPSIP, two discontinuous Galerkin, as well as the C0 interior penalty for piecewise quadratic polynomials and the more involved fifth-order Argyris finite element methods. The role of a smoother in the source as well in the nonlinearities is discussed for the streamline-vorticity formulation of the of incompressible 2D Navier-Stokes equations and the von-Karman plate problems as well as for the biharmonic eigenvalue problem.

The adaptive variants of the Morley and Argyris approximations are compared and result in the rehabilitation of the high-order techniques.  

The presentation is based on joint work with B. Grassle (University of Zurich) and N. Nataraj (IITB in Mumbai) partly reflected in the references below.

C. Carstensen, N. Nataraj, G.C. Remesan, D. Shylaja: Lowest-order FEM for fourth-order

semi-linear problems with trilinear nonlinearity. Numer Math 154 (2023) 323-368.

Carstensen, C. and Grassle B.: Rate-optimal higher-order adaptive conforming FEM for biharmonic eigenvalue problems on polygonal domains. CMAME 425 (2024) 116931

Carstensen, C. and Grassle, B.: Adaptive Morley FEM for 2D stationary Navier-Stokes.  Math Comp (2015) https://doi.org/10.1090/mcom/4069

Efficient and physics-preserving algorithms for thermodynamically consistent flow models in porous media

Huang-xin Chen

Xiamen University, China

Abstracts: Modeling and simulation of two-phase flow and gas flow in porous media are of great interest in the fields of hydrology and petroleum reservoir engineering. In this talk we will introduce a thermodynamically consistent mathematical model for incompressible and immiscible two-phase flow in porous media with rock compressibility. An energy stable numerical method will be introduced, which can preserve multiple physical properties, including the energy dissipation law, full conservation law for both fluids and pore volumes, and bounds of porosity and saturations. Then, the thermodynamically consistent model for multicomponent flow in porous media with rock compressibility and in poroelasticity media will be introduced, and the energy-stable and conservative algorithms will be discussed. Furthermore, we will introduce a natural framework based on the Onsager principle for constructing energy-stable time discretization schemes for multi-phase flow in porous media.

Nonlinear upscaling techniques based on multiscale methods

Eric Chung

The Chinese University of Hong Kong

Abstract: We will present a nonlinear multiscale method in this talk. Our proposed method consists of identifying multi-continua parameters via local basis functions and constructing non-local (in the oversampled region) transfer and effective properties. To achieve this, we significantly modify our recent work proposed within Generalized Multiscale Finite Element Method (GMsFEM) and derive appropriate local problems in oversampled regions once we identify important modes representing each continua. We use piecewise constant functions in each fracture network and in the matrix to write an upscaled equation. Thus, the resulting upscaled equation is of minimal size and the unknowns are average pressures in the fractures and the matrix. Our main contribution is identifying appropriate local problems together with local spectral modes to represent each continua. The model problem for fractures assumes that one can identify fracture networks. The resulting non-local equation (restricted to the oversampling region, which is several times larger compared to the target coarse block). We present numerical results, which show that the proposed approach can provide good accuracy. The research is partially supported by the Hong Kong RGC General Research Fund (Project: 14305624 and 14304021).

The p-Laplace and p-Stokes system

Lars Diening

University of Bielefeld, Germany

Abstract: We present established and recent results on the p-Laplace and the p-Stokes system.  This includes a priori and a posteriori estimates and iterative algorithms to solve these non-linear problems.

A locally conservative proximal Galerkin method for pointwise bound constraints

Guo-sheng Fu

University of Notre Dame, USA

Abstracts: We introduce the first-order system proximal Galerkin (FOSPG) method, a locally mass-conserving, hybridizable finite element method for solving heterogeneous anisotropic diffusion and obstacle problems. Like other proximal Galerkin methods, FOSPG finds solutions by solving a recursive sequence of smooth, discretized, nonlinear subproblems. We establish the well-posedness and convergence of these nonlinear subproblems. Further, we show stability and error estimates under low regularity assumptions for the linearized equations obtained by solving each subproblem using Newton's method.  The FOSPG method exhibits several advantages, including high-order accuracy, discrete maximum principle or bound-preserving discrete solutions, and local mass conservation. It also achieves prescribed solution accuracy within asymptotically mesh-independent numbers of subproblems and linear solves per subproblem iteration. Numerical experiments on benchmarks for anisotropic diffusion and obstacle problems confirm these attributes.

Variational-hemivariational inequalities: theory, numerical analysis, and applications

Wei-min Han

University of Iowa, USA

Abstracts: In recent years, modeling, mathematical analysis, and numerical solution of hemivariational inequalities, or more generally, variational-hemivariational inequalities, have attracted much attention in the research communities. Through the formulation of variational-hemivariational inequalities, application problems involving nonsmooth, monotone or non-monotone, multivalued constitutive laws, forces, and boundary conditions can be treated successfully. Variational-hemivariational inequalities have been successfully applied in a wide variety of subjects, ranging from nonsmooth mechanics, physics, engineering, to economics. 

This talk will provide an overview of the mathematical theory and numerical analysis of families of variational-hemivariational inequalities, with applications in solid mechanics and fluid mechanics.

Computing rough solutions of nonlinear dispersive and wave equations

Bu-yang Li

The Hong Kong Polytechnic University

Abstracts: We report some recent progress on the development of novel numerical methods for computing rough and possibly discontinuous solutions of nonlinear dispersive and wave equations, including the nonlinear wave equation and the KdV equation.

Monotone methods for viscosity solutions of nonlinear PDEs

Wen-bo Li

Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Abstracts: In this talk, we investigate viscosity solutions for a class of nonlinear PDEs, including the Monge-Ampère equation and the infinity Laplace equation. We develop monotone two-scale methods and establish their convergence within the Barles-Souganidis framework. Subsequently, we derive error estimates for solutions with low regularity by leveraging the structures of the problems. Numerical experiments are also presented to validate our theories.

A stabilized P1-P0 finite element method for Navier-Stokes equations

Yu-wen Li

Zhejiang University, China

Abstracts: In this talk, we present a stabilized P1-P0 finite element method for solving the incompressible Navier-Stokes equation. The numerical scheme is based on a reduced Bernardi-Raugel element with statically condensed face bubbles and is pressure robust in the small viscosity regime. For the Navier-Stokes equation, the convection term is discretized using an edge-averaged finite element technique. The material of the talk is based on joint work with Ludmil Zikatanov.

An essentially oscillation-free discontinuous Galerkin method for hyperbolic conservation laws

Yong Liu

Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Abstracts: In this talk, we propose a novel discontinuous Galerkin (DG) method to control the spurious oscillations when solving the hyperbolic conservation laws. The spurious oscillations may be harmful to the numerical simulation, as it not only generates some artificial structures not belonging to the problems but also causes many overshoots and undershoots that make the numerical scheme less robust.

To overcome this difficulty, we introduce a numerical damping term to control spurious oscillations based on the classic DG formulation. Compared to the classic DG method, the proposed DG method still maintains many good properties, such as the extremely local data structure, conservation, L2-boundedness, optimal error estimates, and superconvergence. We also extend our methods to systems of hyperbolic conservation laws.

Entropy inequalities are crucial to the well-posedness of hyperbolic conservation laws, which help to select the physically meaningful one among the infinite many weak solutions. By combining with quadrature-based entropy-stable DG methods, we also developed the entropy-stable OFDG method. For time discretizations, the modified exponential Runge--Kutta method can avoid additional restrictions of time step size due to the numerical damping. Extensive numerical experiments are shown to demonstrate our algorithm is robust and effective. This is a joint work with Jian-fang Lu (SCUT) and Chi-Wang Shu (Brown Univ.).

A Gauss’s law preserving, helicity, mass, charge current, energy-conserving finite element method for MHD systems

Shi-peng Mao

Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Abstracts: We propose a novel structure-preserving finite element scheme for the three-dimensional incompressible magnetohydrodynamic (MHD) equations. The contribution of our research is three-fold.  Firstly, the proposed scheme exactly preserves critical physical properties, including mass conservation, the magnetic Gauss’s law, charge conservation, energy conservation and magnetic/fluid helicity conservation in their respective physical limits. To the best of our knowledge, this is the first numerical method that preserves all these properties simultaneously. Secondly, it introduces the first linear scheme that upholds the helicity preserving property for MHD, thus eliminating the necessity for fixed-point iterations as seen in existing literature. Last but not least, for the resulting large linear systems, we develop efficient block preconditioners that remain robust at high fluid and magnetic Reynolds numbers by incorporating techniques such as the augmented Lagrangian method and mass augmentation. Finally, a series of numerical experiments demonstrate that our method is accurate, stable, robust under extreme physical parameters and capable of preserving all the stated physical properties, including benchmark problems of Orszag Tang vortex and driven magnetic reconnection with fluid and magnetic Reynolds numbers up to 10^6.

Positivity preserving finite element methods for the nonlinear Gross-Pitaevskii eigenvalue problem

Daniel Peterseim

University of Augsburg, Germany

Abstracts: We propose a positivity preserving finite element discretization for the nonlinear Gross-Pitaevskii eigenvalue problem, using mass lumping techniques to ensure that the uniqueness (up to sign) and positivity of the continuous ground state carry over to the discrete setting. We prove that every non-negative discrete excited state coincides (up to sign) with the discrete ground state. This result allows us to characterize the limits of a Riemannian gradient descent method with an energy-adapted metric, thus establishing global convergence to the discrete ground state for the nonlinear finite element eigenvalue problem. In addition, we present an a priori error analysis showing optimal rates of convergence under mesh refinement. Numerical experiments verify our theoretical results.

A nonlinear least-squares convexity enforcing C^0 interior penalty method for the Monge-Ampere equation on strictly convex smooth planar domains

Zhi-yu Tan

Xiamen University, China

Abstracts: We construct a nonlinear least-squares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the Monge-Ampere equation on strictly convex smooth domains in R^2. It is based on an isoparametric C^0 finite element space with exotic degrees of freedom that can enforce the convexity of the approximate solutions. A priori and a posteriori error estimates together with corroborating numerical results are presented.

A parameter-free discontinuous Galerkin method for convex minimization problems

Ngoc Tien Tran

University of Augsburg, Germany

Abstracts: This talk presents an unconditionally stable discontinuous Galerkin method for a class of convex minimization problems. We establish convergence of the method under a two-sided growth condition and derive basic convergence rate estimates.

A posteriori error estimates robust with respect to nonlinearities and orthogonal decomposition based on iterative linearization

Martin Vohralik

Inria Paris & CERMICS, Ecole nationale des ponts et chaussées, France

Abstracts: We discuss a posteriori error estimates for strongly monotone and Lipschitz-continuous nonlinear elliptic problems, where standard approaches do not give estimates robust with respect to the strength of the nonlinearities in the sense that the overestimation factor increases when the problem is more and more nonlinear. We derive estimates that include, and build on, common iterative linearization schemes such as Zarantonello, Picard, Newton, or M- and L-ones. We derive two approaches that give robustness: we either estimate the energy difference that we augment by the discretization error of the current linearization step, or we design iteration-dependent norms that feature weights given by the current linearization iterate. The second setting allows for error localization and an orthogonal decomposition into discretization and linearization components. Numerical experiments illustrate the theoretical findings, with the overestimation factors close to the optimal value of one for any strength of the nonlinearities. Details are given in:

A. Harnist, K. Mitra, A. Rappaport, M. Vohralik, Robust augmented energy a posteriori estimates for Lipschitz and strongly monotone elliptic problems, HAL Preprint~04033438, 2024.

K. Mitra, M. Vohralik, Guaranteed, locally efficient, and robust a posteriori estimates for nonlinear elliptic problems in iteration-dependent norms. An orthogonal decomposition result based on iterative linearization, HAL Preprint~04156711, 2023.

Adaptive finite element method for a nonlinear Helmholtz equation with high wave number

Hai-jun Wu

Nanjing University, China

Abstract: A nonlinear Helmholtz (NLH) equation with high frequencies and corner singularities is discretized by the linear finite element method (FEM). After deriving some wave-number-explicit stability estimates and the singularity decomposition for the NLH problem, a priori stability and error estimates are established for the FEM on shape regular meshes including the case of locally refined meshes. Then a posteriori upper and lower bounds using a new residual-type error estimator, which is equivalent to the standard one, are derived for the FE solutions to the NLH problem. These a posteriori estimates have confirmed a significant fact that is also valid for the NLH problem, namely the residual-type estimator seriously underestimates the error of the FE solution in the preasymptotic regime, which was first observed by Babuska et al. [Int J Numer Methods Eng 40 (1997)] for a one-dimensional linear problem. Based on the new a posteriori error estimator, both the convergence and the quasi-optimality of the resulting adaptive finite element algorithm are proved the first time

for the NLH problem, when the initial mesh size lying in the preasymptotic regime. Finally, numerical examples are presented to validate the theoretical findings and demonstrate that applying the continuous interior penalty (CIP) technique with appropriate penalty parameters can reduce the pollution errors efficiently. In particular, the nonlinear phenomenon of optical bistability with Gaussian incident waves is successfully simulated by the adaptive CIPFEM.

A stabilized nonconforming finite element method for the surface biharmonic problem

Shuo-nan Wu

Peking University, China

Abstracts: This talk presents a novel stabilized nonconforming finite element method for solving the surface biharmonic problem. The method extends the New-Zienkiewicz-type (NZT) element to polyhedral (approximated) surfaces by employing the Piola transform to define the connection of vertex gradients between adjacent elements. Key features of the surface NZT finite element space include its H1-relative conformity and the weak H(div) conformity of the surface gradient, allowing for stabilization without the need for artificial parameters. Assuming that the exact solution and the dual problem possess only H3 regularity, we establish optimal error estimates in the energy norm and provide, for the first time, a detailed analysis yielding optimal second-order convergence in the broken H1 norm. Numerical experiments are provided to support the theoretical results, and they suggest that the stabilization term may not be necessary.

Low-degree finite element methods for transmission eigenvalue problems

Shuo Zhang

Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Abstracts: The transmission eigenvalue problem is applied in invisibility cloaking, seismic exploration, photonic crystals and metasurfaces. As a fourth order nonlinear eigenvalue problem with variable coefficient, it is an important model problem in numerical mathematics. Generally high-degree polynomials are needed to implement a conforming finite element discretization for it. In this talk, we utilize the nonconforming approach and the mixed approach to reduce the polynomial degree.

Particularly, two nonconforming finite element spaces with respectively piecewise cubic (P_3) and quadratic (P_2) polynomials are presented. As a main ingredient, on both spaces, a discrete Grisvard's identity is proved that

\begin{equation}

\sum \int_T\Delta w_h\Delta v_h=\sum\int_T \nabla^2 w_h\nabla^2 v_h,

\end{equation}

which makes the bilinear form $\sum \int_T\alpha(x)\Delta w_h\Delta v_h$ coercive with respect to broken H^2 norm where \alpha is a positive variable coefficient and makes the finite element functions useful for many applications. The two finite element spaces do not conform with the standard definition of ``finite element" by Ciarlet; though, they are implementable by figuring out the basis functions with local supports.

Then by constructing an equivalent stable mixed formulation for the model problem, a mixed element scheme with low degree finite element spaces is easily obtained. The strategy works for the Helmholtz and Maxwell transeigenvalue problems in 2 and 3 dimensions. Further, an optimal multigrid algorithm can be constructed based on the mixed scheme, by which the eigenvalues are computed with optimal computational cost.

Optimal and superconvergence error estimates of P^k-DG methods on rectangular meshes for 2D hyperbolic equations

Zhi-min Zhang

Wayne State University, USA

Abstract: Optimal convergence rate of the P^k discontinuous Galerkin (DG) methods under the rectangular mesh for hyperbolic equations is a long-standing unsolved theoretical problem. Until today, the best result is h+1/2 for general rectangular meshes. In 2020, Liu, Shu, and Zhang proved that under a uniform rectangular mesh, the optimal convergence rate is k+1 for linear problems with constant coefficients. For non-constant coefficients, the optimal rate has been proved for k=0,1,2,3. For the nonlinear case, only k=2,3 have been proved. In this work, we prove the k+1 optimal rate of convergence under the uniform rectangular mesh for variable coefficients and nonlinear cases for all k. In addition, we discover some super-convergence phenomena for P^k-DG for the first time.

Numerical analysis for parameter identification in PDEs

Zhi Zhou

The Hong Kong Polytechnic University

Abstracts: Identifying parameters in partial differential equations (PDEs) represent a very broad class of applied inverse problems. Usually, these problems are addressed through optimization approaches, which are then discretized for practical numerical implementation using finite difference, finite element, or neural network approximations, with the latter often referred to as unsupervised learning in this context. A key challenge in this context is deriving a priori error estimates for the numerical reconstruction of the target parameter. In this talk, we present our recent work on establishing convergence rates for finite element methods in recovering a diffusion coefficient in an elliptic equation. This is achieved by carefully exploiting relevant stability results. Moreover, the approach can be extended to unsupervised learning methods using fully connected neural networks, as well as to multi-parameter identification problems with applications in hybrid physics imaging.