Abstract : Nonlinear partial differential equations (PDEs) have been widely used in various fields, such as thermodynamics, biology, material science, electromagnetism, to name just a few. Even though the history of the study on numerical PDEs is quite long, there are still many open and important questions. In this mini-symposium, we aim at gathering researchers working on the topic to discuss recent advances on the development and numerical analysis of high-order numerical methods for approximately solving nonlinear PDEs, in order to further promote the developments of the topic.
[05134] Error Analysis of IMEX and Time-Splitting Schemes for the Logarithmic Schrodinger’s Equation
Format : Talk at Waseda University
Author(s) :
Li-Lian Wang (Division of Mathematical Sciences, Nanyang Technological University)
Abstract : The Schrodinger’s equation with a logarithmic nonlinear term (LogSE): f(u)=u log(|u|^2) exhibits rich dynamics, but such a nonlinearity presents significant challenges in both numerical solution and error analysis. Compared with usual cubic case, f(u) is non-differentiable at u=0 but possesses certain Holder continuity. In this talk, we shall report our recent attempts in numerical study of LogSE with a focus on time discretization via implicit-explicit scheme and time-splitting scheme and on the introduction of new tools for the error analysis. This talk is based on joint works with Jingye Yan (Jiangsu University, China) and Xiaolong Zhang (Hunan Normal University, China).
[02750] Constructing structure-preserving schemes via Lagrange multiplier approach
Format : Talk at Waseda University
Author(s) :
Qing Cheng (Tongji University)
Jie Shen (Purdue University)
Abstract : In the talk, I will introduce a new Lagrange multiplier approach to construct efficient and accurate structure-preserving schemes for a class of semi-linear and quasi-linear parabolic equations. To be more specific, I will introduce how to construct positivity/bound-preserving, length-preserving, energy-dissipative schemes for a large class of PDEs. I will establish stability results under a general setting, and carry out an error analysis for second-order structure-preserving schemes. Finally, I will apply our approach to several typical PDEs which preserve structures described above. Some numerical results will be presented to validate our approach.
[04415] Optimal $L^2$ error estimates of unconditionally stable FE schemes for the Cahn-Hilliard-Navier-Stokes system
Format : Talk at Waseda University
Author(s) :
Wentao Cai (Beijing Computational Science Research Center)
Weiwei Sun (BNU-HKBU United International College)
Jilu Wang (Harbin Institute of Technology (Shenzhen))
Zongze Yang (The Hong Kong Polytechnic University)
Abstract : The paper is concerned with the analysis of a popular convex-splitting finite element method for the Cahn-Hilliard-Navier-Stokes system, which has been widely used in practice. Since the method is based on a combined approximation to multiple variables involved in the system, the approximation to one of the variables may seriously affect the accuracy for others. Optimal-order error analysis for such combined approximations is challenging. The previous works failed to present optimal error analysis in $L^2$-norm due to the weakness of the traditional approach. Here we first present an optimal error estimate in $L^2$-norm for the convex-splitting FEMs. We also show that optimal error estimates in the traditional (interpolation) sense may not always hold for all components in the coupled system due to the nature of the pollution/influence from lower-order approximations. Our analysis is based on two newly introduced elliptic quasi-projections and the superconvergence of negative norm estimates for the corresponding projection errors. Numerical examples are also presented to illustrate our theoretical results. More important is that our approach can be extended to many other FEMs and other strongly coupled phase field models to obtain optimal error estimates.
[05204] An $L^1$ mixed DG method for second-order Elliptic Equations in the Non-divergence Form
Format : Talk at Waseda University
Author(s) :
Weifeng Qiu (City University of Hong Kong)
Jin Ren (Old Dominion University)
Ke Shi (Old Dominion University)
Yuesheng Xu (Old Dominion University)
Abstract : In this talk we present an $L^1$ mixed DG method for second-order elliptic equations in the non-divergence form. The elliptic PDE in nondivergence form arises in the linearization of fully nonlinear PDEs. Due to the nature of the equations, classical finite element methods based on variational forms can not be employed directly. In this work, we propose a new optimization based finite element method which combines the classical DG framework with recently developed $L^1$ optimization technique. Convergence analysis in both energy norm and $L^{\infty}$ norm are obtained under weak regularity assumption of the PDE ($H^1$). Such $L^1$ optimization problems are nondifferentiable and invalidate traditional graidnet methods. To overcome this difficulty, we characterize solutions of $L^1$ optimization as fixed-points of proximity equations and utilize matrix splitting technique to obtain a class of fixed-point proximity algorithms with convergence analysis. In addition, various numerical examples will be displayed to validate the analysis in the end.
[02310] New analysis of a mixed FEM for Ginzburg-Landau Equations
Format : Talk at Waseda University
Author(s) :
Huadong Gao (Huazhong University of Science and Technology)
Abstract : This talk is concerned with new error analysis of a lowest-order backward Euler Galerkin-mixed finite element method for the time-dependent Ginzburg-Landau equations. The method is based on a commonly-used non-uniform approximations $(P_1, ND_1 \times RT_0)$, which has been widely used. We establish the second-order accuracy for the order parameter in spatial direction, although the accuracy for $(\mathbf{curl} \mathbf{A}, \mathbf{A})$ is in the first order only. Our numerical experiments confirm the optimal convergence of $\psi_h$.
[01989] Spectral analysis of a mixed method for linear elasticity
Format : Talk at Waseda University
Author(s) :
Xiang Zhong (City University of Hong Kong)
Weifeng Qiu (City University of Hong Kong)
Abstract : We consider a mixed method for linear elasticity eigenvalue problem,
which approximates numerically the stress, displacement, and rotation, by piecewise (k+1),
k and (k+1)-th degree polynomials (k>=1) on Hseih-Clough-Toucher grids. The numerical eigenfunction
of stress is symmetric. By the discrete H^1-stability of numerical displacement, we prove an O(h^(k+2)) approximation
to the L^2-orthogonal projection of the eigenspace of exact displacement for the
eigenvalue problem, with proper regularity assumption. We also prove that numerical approximation to the
eigenfunction of stress is locking free with respect to Poisson ratio. We introduce a hybridization to reduce
the mixed method to a condensed eigenproblem and prove an O(h^2) initial approximation of the eigenvalue by using the discrete H^1-stability of numerical displacement.
[05392] Pointwise-in-time a posteriori error control for higher-order discretizations of time-fractional parabolic equations
Format : Talk at Waseda University
Author(s) :
Natalia Kopteva
Sebastian Franz (Technical University Dresden)
Abstract : Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in [N. Kopteva, Pointwise-in-time a posteriori error control for time-fractional parabolic equations, Appl. Math. Lett., 123 (2022)]. We improve the earlier time stepping algorithm based on this theory, and specifically address its stable and efficient implementation in the context of high-order methods. The considered methods include an L1-2 method and continuous collocation methods of arbitrary order, for which adaptive temporal meshes are shown to yield optimal convergence rates in the presence of solution singularities.
[02942] Optimal convergence of the arbitrary Lagrangian-Eulerian second-order projection method for the Navier-Stokes equations on an evolving domain
Format : Talk at Waseda University
Author(s) :
Buyang Li (The Hong Kong Polytechnic University)
Qiqi Rao (The Hong Kong Polytechnic University)
Yupei Xie (The Hong Kong Polytechnic University)
Abstract : In this talk, we introduce how to prove the optimal convergence of the arbitrary Lagrangian-Eulerian second-order projection method for the Navier-Stokes equations on an evolving domain.
[03545] Exponential Spactral Method for Semilinear Subdiffusion Equations with Rough Data
Format : Talk at Waseda University
Author(s) :
Qiqi RAO (PolyU)
Abstract : A new spectral method is constructed for the linear and semilinear subdiffusion equations with possibly discontinuous rough initial data. The new method effectively combines several computational techniques, including the contour integral representation of the solutions, the quadrature approximation of contour integrals, the exponential integrator using the de la Vall ́ee Poussin means of the source function, and a decomposition of the time interval geometrically refined towards the singularity of the solution and the source function. Rigorous error analysis shows that the proposed method has spectral convergence for the linear and semilinear subdiffusion equations with bounded measurable initial data and possibly singular source functions under the natural regularity of the solutions.
[05520] A convergent algorithm for the interaction of mean curvature flow and surface diffusion
Format : Online Talk on Zoom
Author(s) :
Charles M. Elliott (University of Warwick)
Harald Garcke (University of Regensburg)
Balázs Kovács (Paderborn University)
Abstract : In this talk we will discuss a numerical approach for the interaction of mean curvature flow and a diffusion process on the surface.
The evolving surface finite element discretisation is analysed for a coupled geometric PDE system.
We will present an algorithm based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface; give insight into the stability estimates; which lead to optimal-order $H^1$-norm error estimates.
We will present numerical experiments reporting on: convergemce, preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.
Based on a joint work with C. M. Elliott (Warwick) and H. Garcke (Regensburg).
