• Abstract

In this paper, we consider the global well-posedness of solutions to a parabolic-parabolic Keller-Segel model with $p$-Laplace diffusion. We first establish a critical exponent $p^∗=3N/(N+1)$ and prove that when $p> p^∗,$ the solution exists globally for arbitrary large initial value. When $1<p≤p^∗,$ there exists an uniformly bounded global strong solution for small initial value, and the solution decays to zero as $t→ ∞.$ This paper improves and expands the results of [Cong and Liu, Kinet. Relat. Models, 9(4), 2016], in which the parabolic-elliptic case is studied.

• General Summary

• Abstract

In scientific computing, traditional numerical methods for partial differential equations (PDEs), such as finite difference method and finite element method, often need to solve (large-scale) linear systems of equations. It is known that classical iterative solvers, such as Jacobi iteration and Gauss-Seidel iteration, have the smoothing property, i.e. the high-frequency part of the solution can be efficiently captured while the low-frequency part cannot. Multigrid offers a general methodology that utilizes the smoothing property of iterative solvers in a hierarchical manner. Meanwhile, machine learning-based methods for PDEs, such as deep operator network and Fourier neural operator, show the spectral bias, i.e. the low-frequency part of the solution can be efficiently captured while the high-frequency part cannot. The recently developed hybrid iterative numerical transferable solver (HINTS) offers an alternative choice that combines the advantages of classical iterative solvers on fine grids and operator learning methods on coarse grids. In this work, we propose a label-free HINTS for PDEs with the following features: (1) the training of the operator learning component is totally label-free, i.e. we do not need solutions to a given problem, which are typically obtained by classical solvers, (2) the resolution of the operator learning component is far coarser than that of the linear system of equations to be solved, (3) the success of label-free HINTS depends on whether the high-frequency component of the solution is captured on fine grids or not. Numerical experiments, including Possion equation in two and three dimensions, Hemholtz equation in two and three dimensions, anisotropic diffusion equation in two dimensions, are conducted to demonstrate the features of the proposed method. Based on these results, we conclude that the label-free HINTS provides a valuable addition for solving linear systems of equations arising from numerical PDEs.

• General Summary

• Abstract

In this paper, we propose a novel high order unfitted finite element method on Cartesian meshes for solving the acoustic wave equation with discontinuous coefficients having complex interface geometry. The unfitted finite element method does not require any penalty to achieve optimal convergence. We also introduce a new explicit time discretization method for the ordinary differential equation (ODE) system resulting from the spatial discretization of the wave equation. The strong stability and optimal $hp$-version error estimates both in time and space are established. Numerical examples confirm our theoretical results.

• General Summary

• Abstract

In this paper, a novel structure-preserving scheme is proposed for solving the three-dimensional Maxwell’s equations. The proposed scheme can preserve all of the desired structures of the Maxwell’s equations numerically, including five energy conservation laws, two divergence-free fields, three momentum conservation laws and a symplectic conservation law. Firstly, the spatial derivatives of the Maxwell’s equations are approximated with Fourier pseudo-spectral methods. The resulting ordinary differential equations are cast into a canonical Hamiltonian system. Then, the fully discrete structure-preserving scheme is derived by integrating the Hamiltonian system using a sixth order average vector field method. Subsequently, an optimal error estimate is established based on the energy method, which demonstrates that the proposed scheme is of sixth order accuracy in time and spectral accuracy in space in the discrete $L^2$-norm. The constant in the error estimate is proved to be only $\mathcal{O}(T),$ where $T > 0$ is the time period. Furthermore, its numerical dispersion relation is analyzed in detail, and a customized fast solver is presented to efficiently solve the resulting discrete linear equations. Finally, numerical results are presented to validate our theoretical analysis.

• General Summary

• Abstract

An inexact framework of the Newton-based matrix splitting (INMS) iterative method is developed to solve the generalized absolute value equation, whose exact version was proposed by Zhou, Wu and Li [ J. Comput. Appl. Math., 394, 2021]. Global linear convergence of the INMS iterative method is investigated in detail. Some numerical results are given to show the superiority of the INMS iterative method.

• General Summary