Special Session 30: Recent Development in Advanced Numerical Methods for Partial Differential Equations

Stochastic operator learning and its applications in uncertainty quantification

Ling Guo Shanghai Normal University Peoples Rep of China Co-Author(s):

Abstract: The traditional DeepONet framework has limitations stemming from the structure of its branch network. In this talk, we will introduce a new, flexible operator learning model designed to accommodate incomplete observations and address the uncertainties that arise in predictions.

Shearlet Scattering Transform and Its Applications

Wei Guo Hebei Normal University Peoples Rep of China Co-Author(s):    Wei Guo

Abstract: Convolutional neural networks have achieved significant success in in signal processing and computer vision, but its underlying mechanisms are not well understood. Recently, the understanding of convolutional neural networks has received more and more attention. The wavelet scattering transform is the pioneering work presented by Mallat who is one the founders of wavelet analysis. It can be proved that it has the properties of translation invariance and deformation stability. In this talk, we will introduce our proposed shearlet scattering transform. It combines the advantages of scattering transform and shearlet. In addition, we construct a hybrid shearlet scattering network by fusing the shearlet scattering transform with an appropriate convolutional neural network, and apply it to COVID-19 detection and fake news detection tasks, both of which achieve good application performance.

Characteristic block-centered finite difference methods for Darcy-Forchheimer compressible miscible displacement problem

Jian Huang Xiangtan University, CHINA Peoples Rep of China Co-Author(s):

Abstract: In this talk, we present characteristic block-centered finite difference methods for solving the nonlinear Darcy-Forchheimer compressible miscible displacement problem in porous media. The block-centered finite difference method is used to discretize the miscible problem, where the pressure equation is described by the nonlinear Darcy-Forchheimer model, and the transport equation is addressed with the help of the characteristic method. Two-grid methods are developed for the nonlinear system. The nonlinear system is linearized using Newton`s method with a small positive parameter to ensure the differentiability of the nonlinear term in the Darcy-Forchheimer equation. A modified two-grid algorithm is proposed to further reduce the computational cost of the time-dependent problem. The proposed methods are rigorously analyzed, and a priori error estimates are provided for the rates of convergence of the velocity, pressure, concentration, and its flux. Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods by comparing the efficiency with that of other solvers, especially in terms of CPU time.

On the optimal order approximation of the partition of unity finite element method

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

Abstract: In the partition of unity finite element method, the nodal basis of the standard linear Lagrange finite element is multiplied by the $P_k$ polynomial basis to form a local basis of an extended finite element space. Such a space contains the $P_1$ Lagrange element space, but is a proper subspace of the $P_{k+1}$ Lagrange element space on triangular or tetrahedral grids. It is believed that the approximation order of this extended finite element is $k$, in $H^1$-norm, as it was proved in the first paper on the partition of unity, by Babuska and Melenk and this space does not even contain the $C^{0}$-$P_{k}$ space. In this talk, we show surprisingly the approximation order is $k + 1$ in $H^1$-norm. In addition, we extend the method to rectangular/cuboid grids and give a proof to this sharp convergence order. This is a joint work with Shangyou Zhang, University of Delaware.

High order conservative arbitrary Lagrangian-Eulerian schemes for two-dimensional radiation hydrodynamics equations

Nuo Lei Academy of Mathematics and Systems Science, CAS Peoples Rep of China Co-Author(s):

Abstract: Radiation hydrodynamics equations (RHE) refer to the study of how interactions between radiation and matter influence thermodynamic states and dynamic flow, which has been widely applied to high temperature hydrodynamics, such as inertial confinement fusion (ICF). The equations exhibit strong nonlinearity, multi-scale characteristics, and sharp discontinuities, presenting considerable challenges for high-order numerical solutions. To address these, we develop a two-dimensional high-order conservative arbitrary Lagrangian-Eulerian (ALE) scheme. We first design a high-order explicit Lagrangian scheme under the equilibrium diffusion limit based on multi-resolution weighted essentially non-oscillatory (WENO) reconstruction for spatial discretization, strong stability-preserving Runge-Kutta time discretization, and HLLC numerical fluxes, with a focus on discussing the positivity-preserving property of the high-order scheme. In the meantime, to overcome the severe time step restrictions of explicit schemes, we propose a high-order Explicit-Implicit-Null (EIN) Lagrangian scheme by adding linear artificial diffusion terms to the equations, treating nonlinear terms explicitly and handling linear diffusion terms implicitly. Finally, to address the challenges posed by mesh distortion and deformation in Lagrangian methods, we incorporate mesh rezoning and remapping algorithms to develop a high-order conservative ALE scheme suitable for handling complicated RHE. Additionally, we extended the high-order conservative ALE scheme to the non-equilibrium three-temperature RHE. Numerical experiments demonstrate that these schemes are high-order accurate, conservative, non-oscillatory, and can capture the interfaces automatically.

Parametric finite element methods for anisotropic axisymmetric flows

Meng Li Zhengzhou University Peoples Rep of China Co-Author(s):

Abstract: This report considers parametric finite element methods for anisotropic flows in axisymmetric settings, including surface diffusion, conserved mean curvature, power mean curvature and intermediate evolution flows. We introduce novel weak formulations, and different approximating methods are constructed and studied. Computational experiments are presented to demonstrate the efficiency of the proposed methods.

A general collocation analysis for weakly singular Volterra integral equations with variable exponent

Hui Liang Harbin Institute of Technology, Shenzhen Peoples Rep of China Co-Author(s):

Abstract: Piecewise polynomial collocation of weakly singular Volterra integral equations (VIEs) of the second kind has been extensively studied in the literature. Variable-order fractional-derivative differential equations currently attract much research interest, and in Zheng and Wang SIAM J. Numer. Anal. 2020 such a problem is transformed to a weakly singular VIE whose kernel has the variable exponent, then solved numerically by piecewise linear collocation, but it is unclear whether this analysis could be extended to more general problems or to polynomials of higher degree. In the present paper the general theory (existence, uniqueness, regularity of solutions) of variable-exponent weakly singular VIEs is developed, then used to underpin an analysis of collocation methods where piecewise polynomials of any degree can be used. This error analysis is also novel--it makes no use of the usual resolvent representation, which is a key technique in the error analysis of collocation methods for VIEs in the current research literature. Furthermore, all the above analysis for a scalar VIE can be extended to certain nonlinear VIEs and to systems of VIEs. The sharpness of the theoretical error bounds obtained for the collocation methods is demonstrated by numerical examples.

Maximum bound principle and original energy dissipation of arbitrarily high-order rescaled exponential time differencing Runge--Kutta schemes for Allen--Cahn equations

Chaoyu Quan The Chinese University of Hong Kong (Shenzhen) Peoples Rep of China Co-Author(s):    Xiaoming Wang, Pinzhong Zheng, Zhi Zhou

Abstract: The energy dissipation law and the maximum bound principle are two critical physical properties of the Allen--Cahn equations. While many existing time-stepping methods are known to preserve the energy dissipation law, most apply to a modified form of energy. In this work, we demonstrate that, when the nonlinear term of the Allen--Cahn equation is Lipschitz continuous, a class of arbitrarily high-order exponential time differencing Runge--Kutta (ETDRK) schemes preserve the original energy dissipation property, under a mild step-size constraint. Additionally, we guarantee the Lipschitz condition on the nonlinear term by applying a rescaling post-processing technique, which ensures that the numerical solution unconditionally satisfies the maximum bound principle. Consequently, our proposed schemes maintain both the original energy dissipation law and the maximum bound principle and can achieve arbitrarily high-order accuracy. We also establish an optimal error estimate for the proposed schemes. Some numerical experiments are carried out to verify our theoretical results.

Virtual element methods for elastic shell models

Xiaoqin Shen Xi`an University of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will share our research on numerical approximation for two-dimensional three-component (2D-3C) elastic shell problems. When employing the finite element method to address intricate shell surfaces, we often find ourselves constrained to basic geometric meshes, such as triangles and quadrangles. To fulfill the demands of calculation accuracy, these meshes need to be refined, which significantly increases computational expenses. Therefore, we have delved into exploring the virtual element methods (VEM) for Koiter`s model, the elliptic membrane shell model, and the flexural shell model. These methods boast high mesh flexibility, obviate the need for explicit basis function expressions, and elevate convergence accuracy to any desired order. Furthermore, our novel contributions ensure the well-posedness and stability of the approximate problem and analyze the convergence of the numerical solution. Ultimately, our numerical results demonstrate that the discretization scheme effectively solves shell models, is adaptable to arbitrary mesh partitions, and reduces computational costs compared to traditional finite element methods.

Multiscale Model Reduction for Heterogeneous Perforated Domains based on CEM-GMsFEM

Yin Yang Xiangtan University Peoples Rep of China Co-Author(s):

Abstract: In this presentation, we unveil a robust framework for addressing multiscale complexities in diverse perforated domains, employing the Constraint Energy Minimizing - Generalized Multiscale Finite Element Method (CEM-GMsFEM). Simulating within such domains is computationally intensive due to varying perforation and domain scales. Our method addresses both the Poisson equation and linear elastic problems within these domains. Our approach comprises two main steps: firstly, solving an eigenvalue problem within a coarse block, and secondly, resolving a minimization problem within an oversampled domain. Furthermore, we demonstrate the variability of oversampling layers in controlling exponential decay.

Error estimates of finite element methods for nonlocal problems with exact or approximated interaction neighborhoods

Xiaobo Yin Central China Normal University Peoples Rep of China Co-Author(s):    Qiang Du, Hehu Xie and Jiwei Zhang

Abstract: In this talk, we report our recent study on the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter, the radius of the spherical nonlocal interaction neighborhood of the nonlocal model, and the mesh size simultaneously approach zero. Two important cases are discussed: one involving the original nonlocal models and the other for nonlocal models with polygonal approximations of the nonlocal interaction neighborhood. Results of numerical experiments are also reported to substantiate the theoretical studies.

An Alternating Direction Implicit Method for Mean Curvature Flows

Wenjun Ying Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Han Zhou

Abstract: In this talk, we present a Cartesian grid-based method for solving mean curvature flows in two and three space dimensions. The mean curvature flows describe the dynamics of a hypersurface whose normal velocity is determined by local mean curvature. The proposed method embeds a closed hypersurface into a fixed Cartesian grid and decomposes it into multiple overlapping subsets. For each subset, extra tangential velocities are introduced such that marker points on the hypersurface only moves along grid lines. By utilizing an alternating direction implicit (ADI)-type time integration method, the subsets are evolved alternately by solving scalar parabolic partial differential equations on planar domains. The method removes the stiffness using a semi-implicit scheme and has no high-order stability constraint on time step size. Numerical examples in two and three space dimensions are presented to validate the proposed method.

An operator/direction splitting approach to a class of dissipative systems

Zhen Zhang Southern University of Science and Technology Peoples Rep of China Co-Author(s):    Nan Lu, Chenxi Wang, Zhen Zhang

Abstract: In this work, we present a novel operator/direction splitting approach for numerically solving a class of dissipative systems. The proposed methods enjoy the properties of second-order accuracy, energy stability and computationally efficiency. These are validated for many models including Cahn-Hilliard-Navier-Stokes system, phase-field surfactant system, Poisson-Nernst-Planck model, etc.

Addressing complex boundary conditions of miscible flow and transport with application to optimal control

Xiangcheng Zheng Shandong University Peoples Rep of China Co-Author(s):    Yiqun Li, Hong Wang, Xiangcheng Zheng

Abstract: We investigate complex boundary conditions of the miscible displacement system in two and three space dimensions with the commonly-used Bear-Scheidegger diffusion-dispersion tensor, which describes, e.g., the porous medium flow processes in petroleum reservoir simulation or groundwater contaminant transport. Specifically, we incorporate the no-flux boundary condition for the Darcy velocity to prove that the general no-flux boundary condition for the transport equation is equivalent to the normal derivative boundary condition of the concentration, based on which we further prove several complex boundary conditions by the Bear-Scheidegger tensor and its derivative. The derived boundary conditions provide new insights and properties of the Bear-Scheidegger diffusion-dispersion tensor, facilitate the application of classical methods and results without technical treatments for complex boundary conditions, and accommodate the coupling and the nonlinearity of the miscible displacement system and the Bear-Scheidegger tensor in deriving the first-order optimality condition of the corresponding optimal control problem for practical application.