Special Session 58: Recent Advances in Numerical Methods for Partial Differential Equations

Small cut cells in a fictitious domain approach for fluid structure interactions

Daniele Boffi King Abdullah University of Science and Technology Saudi Arabia Co-Author(s):

Abstract: In this talk I will present recent advances on the numerical analysis and implementation of a fictitious domain approach for the approximation of fluid structure interaction problems. As opposed to other unfitted discretizations, our analysis doesn`t require any particular treatment of small cells generated by elements cut by the interface between fluid and solid. Several numerical experiments confirm the stability of the method independently on the size of the cut cells.

Highly efficient and energy stable multi-step SAV approaches for phase field models

Yanping Chen Nanjing University of Posts and Telecommunications, CHINA Peoples Rep of China Co-Author(s):

Abstract: Recently, the scalar auxiliary variable (SAV) approach and its extended SAV-based approaches have been widely used to simulate a series of phase field models. However, many SAV-based schemes are known for the stability of a modified energy. In this paper, we construct a series of modified SAV approaches with unconditional energy dissipation law based on several improvements to the traditional SAV approach. Firstly, by introducing the three-step technique, we can reduce the number of constant coefficient linear equations that need to be solved at each time step, while retaining all of its other advantages. Secondly, the addition of energy-optimized technique and SAV/Lagrange multiplier technique can make the numerical schemes have the advantage of preserving the original energy dissipation. Thirdly, we use the first-order approximation of the energy balance equation in the GSAV approach, instead of discretizing the dynamic equation of the auxiliary variable, so that we can construct the high-order unconditional original energy stable numerical schemes. Finally, representative numerical examples show that the efficiency and accuracy of the proposed schemes are improved.

A Construction of $C^r$ Conforming Finite Element Spaces in Any Dimension

Jun Hu Peking University Peoples Rep of China Co-Author(s):

Abstract: This talk proposes a construction of $C^r$ conforming finite element spaces with arbitrary $r$ in any dimension. It is shown that if $k \ge 2^{d}r+1$ the space $P_k$ of polynomials of degree $\le k$ can be taken as the shape function space of $C^r$ finite element spaces in $d$ dimensions. This is the first work on constructing such $C^r$ conforming finite elements in any dimension in a unified way.

Green Multigrid Network

Jiwei Jia Jilin University Peoples Rep of China Co-Author(s):    Jiwei Jia, Young Ju Lee, Ye Lin

Abstract: We propose a framework of the Green Multigrid network (GreenMGNet), a type of operator learning algorithm for a class of asymptotically smooth Green functions. The new framework presents itself better accuracy and efficient computational complexity, thereby achieving a significant improvement. GreenMGNet is composed of two technical novelties. First, the Green function is modeled as a piecewise function to preserve its singular behavior in some part of the hyperplane. Such piecewise function is then approximated by a neural network with augmented output(AugNN), so that it can capture singularity accurately. Second, the asymptotic smoothness property of the Green function is used to leverage Multi-Level Multi-Integration(MLMI) algorithm for both training and inference stages. Several test cases of operator learning are presented to demonstrate the accuracy and effectivity of the proposed method. On average, GreenMGNet achieves 3.8% to 39.15% accuracy improvement. To match the accuracy level of GL, GreenMGNet requires only about 10% of the full grid data, resulting in a 55.9% and 92.5% reduction in training time and GPU memory cost for one-dimensional test problems, and a 37.7% and 62.5% reduction for two-dimensional test problems.

A perfectly matched layer method for scattering problem in cylindrical coordinates

Xue Jiang Beijing University of Technology Peoples Rep of China Co-Author(s):

Abstract: This talk is focused on the modelling of signal propagations in myelinated axons. The well-posedness of model is established upon Dirichlet boundary conditions at the two ends of the neural structure and the radiative condition in the radial direction of the structure. Using the perfectly matched layer (PML) method, we truncate the unbounded background medium and propose an approximate problem on the truncated domain. The well-posedness of the PML problem and the exponential convergence of the approximate solution to the exact solution are established. Numerical experiments are presented to demonstrate the theoretical results and the efficiency of our methods to simulate the signal propagation in axons.

A hybrid iterative method based on MIONet for PDEs: Theory and numerical examples

Pengzhan Jin Peking University Peoples Rep of China Co-Author(s):

Abstract: We propose a hybrid iterative method based on MIONet for PDEs, which combines the traditional numerical iterative solver and the recent powerful machine learning method of neural operator, and further systematically analyze its theoretical properties, including the convergence condition, the spectral behavior, as well as the convergence rate, in terms of the errors of the discretization and the model inference. We show the theoretical results for the frequently-used smoothers, i.e. Richardson (damped Jacobi) and Gauss-Seidel. We give an upper bound of the convergence rate of the hybrid method w.r.t. the model correction period, which indicates a minimum point to make the hybrid iteration converge fastest. Several numerical examples including the hybrid Richardson (Gauss-Seidel) iteration for the 1-d (2-d) Poisson equation are presented to verify our theoretical results, and also reflect an excellent acceleration effect. As a meshless acceleration method, it is provided with enormous potentials for practice applications.

Efficient quantum Gibbs samplers

Bowen Li City University of Hong Kong Hong Kong Co-Author(s):

Abstract: Lindblad dynamics and other open-system dynamics provide a promising path towards efficient Gibbs sampling on quantum computers. In these proposals, the Lindbladian is obtained via an algorithmic construction akin to designing an artificial thermostat in classical Monte Carlo or molecular dynamics methods, rather than treated as an approximation to weakly coupled system-bath unitary dynamics. In this talk, we build upon the structural characterization of KMS detailed balanced Lindbladians by Fagnola and Umanita, and develop a family of efficient quantum Gibbs samplers using a finite set of jump operators (the number can be as few as one), akin to the classical Markov chain-based sampling algorithm. Compared to the existing works, our quantum Gibbs samplers have a comparable quantum simulation cost but with greater design flexibility and a much simpler implementation and error analysis. In addition, we will present an efficient preparation of low temperature Gibbs state for 2D toric code by an improved mixing time analysis.

DG method for fractional Laplace equations

Wenbo Li The Academy of Mathematics and Systems Science of the Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Juan Pablo Borthagaray

Abstract: In this talk we consider the integral fractional Laplace equation on bounded domains. We first review the basic theories including the regularity of solutions and convergence rates of standard conforming finite element method. Next, we introduce a ``DG`` formulation motivated by an integration by parts formula for fractional Laplacian and establish the well-posedness of this discretization. We also derive the convergence rates and justify its optimality by some numerical experiments. Some variants of the bilinear form and hanging nodes on the shape-regular mesh are also permitted in our theory. In the end, we apply this idea to the problem of fractional Laplacian with order higher than one.

An Energy-stable Numerical Approximation for the Willmore Flow

Yifei Li Tuebingen University Peoples Rep of China Co-Author(s):    Weizhu Bao

Abstract: The Willmore energy has widespread applications in differential geometry, cell membranes, optical lenses, materials science, among others. The Willmore flow, as the $L^2$ gradient flow dissipating the Willmore energy, serves as a fundamental tool for its analysis. Despite its importance, the development of energy-stable parametric methods for the Willmore flow remains open. In this talk, we present a novel energy-stable numerical approximation for the Willmore flow. We begin by introducing our method for planar curves, then demonstrating the underlying ideas -- the new transport equation and the time derivative of the mean curvature, that ensure energy stability. Finally, we discuss the extension of our approach to surfaces in 3D.

High accuracy algorithm and analysis for nonconforming element of Stokes equation

Limin Ma Wuhan University Peoples Rep of China Co-Author(s):

Abstract: For the Crouzeix-Raviart and enriched Crouzeix-Raviart elements of the Stokes problem, two pseudostress interpolations are designed and proved to admit a full one-order supercloseness with respect to the numerical velocity and the pressure, respectively. The design of these interpolations overcomes the difficulty caused by the lack of supercloseness of the canonical interpolations for the two nonconforming elements, and leads to an intrinsic and concise asymptotic analysis of numerical eigenvalues for the Stokes operator, which proves an optimal superconvergence of eigenvalues by the extrapolation algorithm. Meanwhile, an optimal superconvergence of postprocessed approximations for the Stokes equation is proved by use of this supercloseness. Finally, numerical experiments are tested to verify the theoretical results.

The condition for constructing a finite element from a superspline

Qingyu Wu Peking University Peoples Rep of China Co-Author(s):    Jun Hu, Ting Lin, Beihui Yuan

Abstract: This talk addresses the sufficient and necessary conditions for constructing $C^r$ conforming finite element spaces from a superspline spaces on general simplicial triangulations. We introduce the concept of extendability for the pre-element spaces, which encompasses both the superspline space and the finite element space. By examining the extendability condition for both types of spaces, we provide an answer to the conditions regarding the construction. A corollary of our results is that constructing $C^r$ conforming elements in $d$ dimensions should in general require an extra $C^{2^{s}r}$ continuity on $s$-codimensional simplices, and the polynomial degree is at least $(2^d r + 1)$.

Weak Galerkin Finite Element Scheme and Its Applications

Ran Zhang Jilin University Peoples Rep of China Co-Author(s):

Abstract: The weak Galerkin (WG) finite element method is a newly developed and efficient numerical technique for solving partial differential equations (PDEs). It was first introduced and analyzed for second order elliptic equations and further applied to several other model equations, such as the Brinkman equations, the eigenvalue problem of PDEs to demonstrate its power and efficiency as an emerging new numerical method. This talk introduces some progress on the WG scheme, which includes the applications on Brinkman problems, etc.

New error analysis of a class of fully discrete finite element methods for the dynamical inductionless MHD equations

Xiaodi Zhang Zhengzhou University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we present a new error analysis of a class of fully discrete finite element methods for the dynamical inductionless magnetohydrodynamic equations. The methods use the semi- implicit backward Euler scheme in time and use the standard inf-sup stable Mini/Taylor-Hood pairs to discretize the velocity and pressure, and the Raviart-Thomas for solving the current density in space. Due to the strong coupling of the system and the pollution of the lower-order Raviart-Thomas face approximation in analysis, the existing analysis is not optimal. In terms of a mixed Poisson projection and the corresponding estimates in negative norms, we establish new and optimal error estimates for all variables. Numerical experiments are performed to verify the theoretical analysis.

A new p-multigrid method for elliptic problems

Weying Zheng Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Nuo Lei and Donghang Zhang

Abstract: In this talk, I will present a new p-multigrid method for solving second-order elliptic equations on structured meshes. Using Gauss-Seidel iterations for both pre- and post-smoothings, we prove the uniform convergence of W-cycle multigrid method with respect to both the mesh size h and the degree of polynomials p, provided that the number of smoothing steps is comparable to the degree of polynomials on each level. The p-multigrid method is robust to high-order polynomials and discontinuous coefficients with large jumps.