• 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.

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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.

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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.

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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.

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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.

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In this work, we consider an inverse potential problem in the parabolic equation, where the unknown potential is a space-dependent function and the used measurement is the final time data. The unknown potential in this inverse problem is parameterized by deep neural networks (DNNs) for the reconstruction scheme. First, the uniqueness of the inverse problem is proved under some regularities assumption on the input sources. Then we propose a new loss function with regularization terms depending on the derivatives of the residuals for partial differential equations (PDEs) and the measurements. These extra terms effectively induce higher regularity in solutions so that the ill-posedness of the inverse problem can be handled. Moreover, we establish the corresponding generalization error estimates rigorously. Our proofs exploit the conditional stability of the classical linear inverse source problems, and the mollification on the noisy measurement data which is set to reduce the perturbation errors. Finally, the numerical algorithm and some numerical results are provided.

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This paper revisits the convergence and convergence rate of the parallel splitting augmented Lagrangian method, which can be used to efficiently solve the separable multi-block convex minimization problem with linear constraints. To make use of the separable structure, the augmented Lagrangian method with Jacobian-based decomposition fully exploits the properties of each function in the objective, and results in easier subproblems. The subproblems of the method can be solved and updated in parallel, thereby enhancing computational efficiency and speeding up the convergence. We further study the parallel splitting augmented Lagrangian method with a modified correction step, which shows improved performance with larger step sizes in the correction step. By introducing a refined correction step size with a tight bound for the constant step size, we establish the global convergence of the iterates and $\mathcal{O}(1/N)$ convergence rate in both the ergodic and non-ergodic senses for the new algorithm, where $N$ denotes the iteration numbers. Moreover, we demonstrate the applicability and promising efficiency of the method with tight step size through some applications in image processing.

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This paper is concerned with the numerical analyses of finite element methods for the nonlinear backward stochastic Stokes equations (BSSEs) where the forcing term is coupled with $z.$ Under several developed analysis techniques, the error estimates of the proposed semi-discrete and fully discrete schemes, as well as their boundedness, are rigorously presented and established. Optimal convergence rates of the fully discrete scheme are obtained not only for the velocity $u$ and auxiliary stochastic process $z$ but also for the pressure $p.$ For the efficiency of solving BSSEs, the proposed numerical scheme is parallelly designed in stochastic space. Numerical results are finally provided and tested in parallel to validate the theoretical results.

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Since the memory effect is taken into account, the singularly perturbed subdiffusion equation can better describe the diffusion phenomenon with small diffusion coefficients. However, near the boundary configured with non-smooth boundary values, the solution of the singularly perturbed subdiffusion equation has a boundary layer of thickness $\mathcal{O}(ε),$ which brings great challenges to the construction of the efficient numerical schemes. By decomposing the Caputo fractional derivative, the singularly perturbed subdiffusion equation is formally transformed into a class of steadystate diffusive-reaction equation. By means of a kind of tailored finite point method (TFPM) scheme for solving steady-state diffusion-reaction equations and the $\mathcal{L}1$ formula for discretizing the Caputo fractional derivative, we construct a new $\mathcal{L}1$-TFPM scheme for solving singularly perturbed subdiffusion equations. Our proposed numerical scheme satisfies the discrete extremum principle and is unconditionally numerically stable. Besides, we prove that the new TFPM scheme can obtain reliable numerical solutions as $h ≪ ε$ and $ε ≪ h.$ However, there will be a large error loss due to the resonance effect as $h ∼ ε.$ Numerical experimental results can demonstrate the validity of the numerical scheme.

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