In this paper, we consider the numerical solution of decoupled mean-field forward backward stochastic differential equations with jumps (MFBSDEJs). By using finite difference approximations and the Gaussian quadrature rule, and the weak order 2.0 Itô-Taylor scheme to solve the forward mean-field SDEs with jumps, we propose a new second order scheme for MFBSDEJs. The proposed scheme allows an easy implementation. Some numerical experiments are carried out to demonstrate the stability, the effectiveness and the second order accuracy of the scheme.

This paper presents various acceleration techniques tailored for the traditional 3D topology optimization problem. Firstly, the adoption of the finite difference method leads to a sparser stiffness matrix, resulting in more efficient matrix-vector multiplication. Additionally, a fully matrix-free technique is proposed, which only assembles stiffness matrices at the coarsest grid level and does not require complex node numbering. Moreover, an innovative N-cycle multigrid (MG) algorithm is proposed to act as a preconditioner within conjugate gradient (CG) iterations. Finally, to further enhance the optimization process on high-resolution grids, a progressive strategy is implemented. The numerical results confirm that these acceleration techniques are not only efficient, but also capable of achieving lower compliance and reducing memory consumption. MATLAB codes complementing the article can be downloaded from Github.

Stochastic gradient descent (SGD) methods have gained widespread popularity for solving large-scale optimization problems. However, the inherent variance in SGD often leads to slow convergence rates. We introduce a family of unbiased stochastic gradient estimators that encompasses existing estimators from the literature and identify a gradient estimator that not only maintains unbiasedness but also achieves minimal variance. Compared with the existing estimator used in SGD algorithms, the proposed estimator demonstrates a significant reduction in variance. By utilizing this stochastic gradient estimator to approximate the full gradient, we propose two mini-batch stochastic conjugate gradient algorithms with minimal variance. Under the assumptions of strong convexity and smoothness on the objective function, we prove that the two algorithms achieve linear convergence rates. Numerical experiments validate the effectiveness of the proposed gradient estimator in reducing variance and demonstrate that the two stochastic conjugate gradient algorithms exhibit accelerated convergence rates and enhanced stability.

In this paper, we present quantum algorithms for a class of highly-oscillatory transport equations, which arise in semi-classical computation of surface hopping problems and other related non-adiabatic quantum dynamics, based on the Born-Oppenheimer approximation. Our method relies on the classical nonlinear geometric optics method, and the recently developed Schrödingerisation approach for quantum simulation of partial differential equations. The Schrödingerisation technique can transform any linear ordinary and partial differential equations into Hamiltonian systems evolving under unitary dynamics, via a warped phase transformation that maps these equations to one higher dimension. We study possible paths for better recoveries of the solution to the original problem by shifting the bad eigenvalues in the Schrödingerized system. Our method ensures the uniform error estimates independent of the wave length, thus allowing numerical accuracy, in maximum norm, even without numerically resolving the physical oscillations. Various numerical experiments are performed to demonstrate the validity of this approach.

The recently developed DOC kernels technique has been successful in the stability and convergence analysis for variable time-step BDF2 schemes. However, it may not be readily applicable to problems exhibiting an initial singularity. In the numerical simulations of solutions with initial singularity, variable time-step schemes like the graded mesh are always adopted to achieve the optimal convergence, whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied. In this paper, we revisit the variable time-step implicit-explicit two-step backward differentiation formula (IMEX BDF2) scheme to solve the parabolic integro-differential equations (PIDEs) with initial singularity. We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios $r_k:=\tau_k / \tau_{k-1}<r_{\text {max }}=4.8645(k \geq 3)$ and a much mild requirement on the first ratio, i.e. $r_2>0$. This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy, i.e. the graded mesh $t_k=T(k / N)^\gamma$. In this situation, the convergence order of $\mathcal{O}\left(N^{-\min \{2, \gamma \alpha\}}\right)$ is achieved, where $N$ denotes the total number of mesh points and $\alpha$ indicates the regularity of the exact solution. This is, the optimal convergence will be achieved by taking $\gamma_{\text {opt }}=2 / \alpha$. Numerical examples are provided to demonstrate our theoretical analysis.

In this paper, we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations (PDEs). The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion. This map takes input of the element geometry and the PDE’s parameters on that element, and gives output of two operators: (1) the in2out operator for inter-element communication, and (2) the in2sol operator (Green’s function) for element-wise solution recovery. A significant advantage of this approach is that, once trained, this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining. Also, the training is significantly simpler since it is done on the element level instead on the entire domain. We call this approach element learning. This method is closely related to hybridizable discontinuous Galerkin (HDG) methods in the sense that the local solvers of HDG are replaced by machine learning approaches. Numerical tests are presented for an example PDE, the radiative transfer or radiation transport equation, in a variety of scenarios with idealized or realistic cloud fields, with smooth or sharp gradient in the cloud boundary transition. Under a fixed accuracy level of 10^-3 in the relative L² error, and polynomial degree p = 6 in each element, we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method.

In this paper, we develop the stabilization-free virtual element method for the Helmholtz transmission eigenvalue problem on anisotropic media. The eigenvalue problem is a variable-coefficient, non-elliptic, non-selfadjoint and nonlinear model. Separating the cases of the index of refraction n ≠ 1 and n ≡ 1, the stabilization-free virtual element schemes are proposed, respectively. Furthermore, we prove the spectral approximation property and error estimates in a unified theoretical framework. Finally, a series of numerical examples are provided to verify the theoretical results, show the benefits of the stabilization-free virtual element method applied to eigenvalue problems, and implement the extensions to high-order and high-dimensional cases.

An explicit numerical method is developed for a class of non-autonomous time-changed stochastic differential equations, whose coefficients obey Hölder’s continuity in terms of the time variables and are allowed to grow super-linearly in terms of the state variables. The strong convergence of the method in the finite time interval is proved and the convergence rate is obtained. Numerical simulations are provided.

This paper constructs the first mixed finite element for the linear elasticity problem in 3D using P₃ polynomials for the stress and discontinuous P₂ polynomials for the displacement on tetrahedral meshes under some mild mesh conditions. The degrees of freedom of the stress space as well as the corresponding nodal basis are established by characterizing a space of certain piecewise constant symmetric matrices on a patch around each edge. Macro-element techniques are used to define a stable interpolation to prove the discrete inf-sup condition. Optimal convergence is obtained theoretically.

In this paper, we propose an accelerated stochastic variance reduction gradient method with a trust-region-like framework, referred as the NMSVRG-TR method. Based on NMSVRG, we incorporate a Katyusha-like acceleration step into the stochastic trust region scheme, which improves the convergence rate of the SVRG methods. Under appropriate assumptions, the linear convergence of the algorithm is provided for strongly convex objective functions. Numerical experiment results show that our algorithm is generally superior to some existing stochastic gradient methods.

In this study, we explore two distinct rational approximations to the radiation condition for effectively solving time-harmonic wave propagation problems governed by the Helmholtz equation in $\mathbb{R}^d$, d = 2 or 3. First, we focus on the well-known complete radiation boundary condition (CRBC), which was developed for a transparent boundary condition for two-dimensional problems. The extension of CRBC to three-dimensional problems is a primary concern. Applications of CRBC require removing a near-cutoff region for a frequency range of a process to minimize reflection errors. To address the limitation faced by the CRBC application we introduce another absorbing boundary condition that avoids this demanding truncation. It is a new rational approximation to the radiation condition, which we call a rational absorbing boundary condition, that is capable of accommodating all types of propagating wave modes, including the grazing modes. This paper presents a comparative performance assessment of two approaches in two and three-dimensional spaces, providing insights into their effectiveness for practical application in wave propagation problems.

An analysis of logistic stochastic differential equations (SDEs) with general power-law and driven by a Wiener process is conducted. We prove existence of unique, strong Markovian, continuous solutions. The solutions live (a.s.) on bounded domains D = [0, K] required by applications to biology, ecology and physics with nonrandom threshold parameter K > 0 (i.e. the maximum carrying constant). Moreover, we present and justify nonstandard numerical methods constructed by specified balanced implicit methods (BIMs). Their weak and L^p-convergence follows from the fact that these methods with local Lipschitz-continuous coefficients of logistic SDEs “produce” positive numerical approximations on bounded domain [0, K] (a.s.). As commonly known, standard numerical methods such as Taylor-type ones for SDEs fail to do that. Finally, asymptotic stability of nontrivial equilibria x_* = K is proven for both continuous time logistic SDEs and discrete time approximations by BIMs. We exploit the technique of positive, sufficiently smooth and Lyapunov functionals governed by well-known Dynkin’s formula for SDEs.

In this paper, we introduce the weak Galerkin (WG) method for solving the coupled Stokes and Darcy-Forchheimer flows problem with the Beavers-Joseph-Saffman interface condition in bounded domains. We define the WG spaces in the polygonal meshes and construct corresponding discrete schemes. We prove the existence and uniqueness of the WG scheme by the discrete inf-sup condition and monotone operator theory. Then, we derive the optimal error estimates for the velocity and pressure. Numerical experiments are presented to verify the efficiency of the WG method.

In this paper, an effective oscillation-free discontinuous Galerkin (DG) scheme for a nonlinear stochastic convection-dominated problem is formulated and analyzed. The proposed oscillation-free scheme is capable to capture the steep fronts of solution automatically and distinguish the influence of the convection domination and noise perturbation. Under proper regularity assumptions, the optimal convergence rates in space and time are rigorously proved with the techniques of variational solution and conditional expectation. In the numerical simulation, the classical SIPG scheme and the proposed oscillation-free DG scheme are both performed and compared. The numerical convergence rates tests are first carried out to verify the theoretical results. The benchmark tests having the steep behaviors are further provided to illustrate the effectiveness and robustness of our proposed oscillation-free DG scheme.

This paper is concerned with the new error analysis of a Hodge-decomposition based finite element method for the time-dependent Ginzburg-Landau equations in superconductivity. In this approach, the original equation of magnetic potential A is replaced by a new system consisting of four scalar variables. As a result, the conventional Lagrange finite element method (FEM) can be applied to problems defined on non-smooth domains. It is known that due to the low regularity of A, conventional FEM, if applied to the original Ginzburg-Landau system directly, may converge to the unphysical solution. The main purpose of this paper is to establish an optimal error estimate for the order parameter in spatial direction, as previous analysis only gave a sub-optimal convergence rate analysis for all three variables due to coupling of variables. The analysis is based on a nonstandard quasi-projection for ψ and the corresponding negative-norm estimate for the classical Ritz projection. Our numerical experiments confirm the optimal convergence of ψ_h.

Anderson acceleration is a kind of effective method for improving the convergence of the general fixed point iteration. In the linear case, Anderson acceleration can be used to improve the convergence rate of matrix splitting based iterative methods. In this paper, by using Anderson acceleration on general splitting iterative methods for linear systems, three classes of methods are given. The first one is obtained by directly applying Anderson acceleration on splitting iterative methods. For the second class of methods, Anderson acceleration is used periodically in the splitting iteration process. The third one is constructed by combining the Anderson acceleration and split iteration method in each iteration process. The key of this class of method is to determine a combination coefficient for Anderson acceleration and split iteration method. One optimal combination coefficient is given. Some theoretical results about the convergence of the considered three methods are established. Numerical experiments show that the proposed methods are effective.

The h-version analysis technique developed in [Banjai et al., SIAM J. Numer. Anal., 55 (2017)] for Trefftz discontinuous Galerkin (DG) discretizations of the second order isotropic wave equation is extended to the time-dependent Maxwell equations in anisotropic media. While the discrete variational formulation and its stability and quasi-optimality are derived parallel to the acoustic wave case, the derivation of error estimates in a mesh-skeleton norm requires new transformation stabilities for the anisotropic case. The error estimates of the approximate solutions with respect to the condition number of the coefficient matrices are proved. Furthermore, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent nonhomogeneous Maxwell equations. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy.

In this work, an unconditionally stable, decoupled, variable time step scheme is presented for the incompressible Navier-Stokes equations. Based on a scalar auxiliary variable in exponential function, this fully discrete scheme combines the backward Euler scheme for temporal discretization with variable time step and a mixed finite element method for spatial discretization, where the nonlinear term is treated explicitly. Moreover, without any restriction on the time step, stability of the proposed scheme is discussed. Besides, error estimate is provided. Finally, some numerical results are presented to illustrate the performances of the considered numerical scheme.

This study proposes a class of augmented subspace schemes for the weak Galerkin (WG) finite element method used to solve eigenvalue problems. The augmented subspace is built with the conforming linear finite element space defined on the coarse mesh and the eigen-function approximations in the WG finite element space defined on the fine mesh. Based on this augmented subspace, solving the eigenvalue problem in the fine WG finite element space can be reduced to the solution of the linear boundary value problem in the same WG finite element space and a low dimensional eigenvalue problem in the augmented subspace. The proposed augmented subspace techniques have the second order convergence rate with respect to the coarse mesh size, as demonstrated by the accompanying error estimates. Finally, a few numerical examples are provided to validate the proposed numerical techniques.

In this paper, we propose a pressure-robust weak Galerkin (WG) finite element scheme to solve the Stokes-Darcy problem. To construct the pressure-robust numerical scheme, we use the divergence-free velocity reconstruction operator to modify the test function on the right side of the numerical scheme. This numerical scheme is easy to implement because it only need to modify the right side. We prove the error between the velocity function and its numerical solution is independent of the pressure function and viscosity coefficient. Moreover, the errors of the velocity function reach the optimal convergence orders under the energy norm, as validated by both theoretical analysis and numerical results.

In this paper, we propose and analyze two second-order accurate finite difference schemes for the one-dimensional heat equation with concentrated capacity on a computational domain $\Omega=[a, b]$. We first transform the target equation into the standard heat equation on the domain excluding the singular point equipped with an inner interface matching (IIM) condition on the singular point $x=\xi \in(a, b)$, then adopt Taylor's expansion to approximate the IIM condition at the singular point and apply second-order finite difference method to approximate the standard heat equation at the nonsingular points. This discrete procedure allows us to choose different grid sizes to partition the two sub-domains $[a, \xi]$ and $[\xi, b]$, which ensures that $x=\xi$ is a grid point, and hence the proposed schemes can be generalized to the heat equation with more than one concentrated capacities. We prove that the two proposed schemes are uniquely solvable. And through in-depth analysis of the local truncation errors, we rigorously prove that the two schemes are second-order accurate both in temporal and spatial directions in the maximum norm without any constraint on the grid ratio. Numerical experiments are carried out to verify our theoretical conclusions.

We analyze the long-time behavior of numerical schemes for a class of monotone stochastic partial differential equations (SPDEs) driven by multiplicative noise. By deriving several time-independent a priori estimates for the numerical solutions, combined with the ergodic theory of Markov processes, we establish the exponential ergodicity of these schemes with a unique invariant measure, respectively. Applying these results to the stochastic Allen-Cahn equation indicates that these schemes always have at least one invariant measure, respectively, and converge strongly to the exact solution with sharp time-independent rates. We also show that these numerical invariant measures are exponentially ergodic and thus give an affirmative answer to a question proposed in [J. Cui et al., Stochastic Process. Appl., 134 (2021)], provided that the interface thickness is not too small.

The main purpose of this paper is to give stability analysis and error estimates of the ultra-weak local discontinuous Galerkin (UWLDG) method coupled with a spectral deferred correction (SDC) temporal discretization method up to fourth order, for solving the fourth-order equation. The UWLDG method introduces fewer auxiliary variables than the local discontinuous Galerkin method and no internal penalty terms are required for stability, which is efficient for high order partial differential equations (PDEs). The SDC method we adopt in this paper is based on second-order time integration methods and the order of accuracy is increased by two for each additional iteration. With the energy techniques, we rigorously prove the fully discrete schemes are unconditionally stable. By the aid of special projections and initial conditions, the optimal error estimates of the fully discrete schemes are obtained. Furthermore, we generalize the analysis to PDEs with higher even-order derivatives. Numerical experiments are displayed to verify the theoretical results.

This paper presents a simple proof for the stability of circular perfectly matched layer (PML) methods for solving acoustic scattering problems in two and three dimensions. The medium function of PML allows arbitrary-order polynomials, and can be extended to general nondecreasing functions with a slight modification of the proof. In the regime of high wavenumbers, the inf-sup constant for the PML truncated problem is shown to be $\mathcal{O}$(k^-1). Moreover, the PML solution converges to the exact solution exponentially, with a wavenumber-explicit rate, as either the thickness or medium property of PML increases. Numerical experiments are presented to verify the theories and performances of PML for variant polynomial degrees.

We study decentralized smooth optimization problems over compact submanifolds. Recasting it as a composite optimization problem, we propose a decentralized DouglasRachford splitting algorithm (DDRS). When the proximal operator of the local loss function does not have a closed-form solution, an inexact version of DDRS (iDDRS), is also presented. Both algorithms rely on careful integration of the nonconvex Douglas-Rachford splitting algorithm with gradient tracking and manifold optimization. We show that our DDRS and iDDRS achieve the convergence rate of $\mathcal{O}$(1/k). The main challenge in the proof is how to handle the nonconvexity of the manifold constraint. To address this issue, we utilize the concept of proximal smoothness for compact submanifolds. This ensures that the projection onto the submanifold exhibits convexity-like properties, which allows us to control the consensus error across agents. Numerical experiments on the principal component analysis are conducted to demonstrate the effectiveness of our decentralized DRS compared with the state-of-the-art ones.

The strong convergence of an explicit full-discrete scheme is investigated for the stochastic Burgers-Huxley equation driven by additive space-time white noise, which possesses both Burgers-type and cubic nonlinearities. To discretize the continuous problem in space, we utilize a spectral Galerkin method. Subsequently, we introduce a nonlinear-tamed exponential integrator scheme, resulting in a fully discrete scheme. Within the framework of semigroup theory, this study provides precise estimations of the Sobolev regularity, L^∞ regularity in space, and Hölder continuity in time for the mild solution, as well as for its semi-discrete and full-discrete approximations. Building upon these results, we establish moment boundedness for the numerical solution and obtain strong convergence rates in both spatial and temporal dimensions. A numerical example is presented to validate the theoretical findings.

This paper is concerned with the superconvergence error estimates of a classical mixed finite element method for a nonlinear parabolic/elliptic coupled thermistor equations. The method is based on a popular combination of the lowest-order rectangular Raviart-Thomas mixed approximation for the electric potential/field ($\phi, \boldsymbol{\theta}$) and the bilinear Lagrange approximation for temperature $u$. In terms of the special properties of these elements above, the superclose error estimates with order $\mathcal{O}\left(h^2\right)$ are obtained firstly for all three components in such a strongly coupled system. Subsequently, the global superconvergence error estimates with order $\mathcal{O}\left(h^2\right)$ are derived through a simple and effective interpolation post-processing technique. As by a product, optimal error estimates are acquired for potential/field and temperature in the order of $\mathcal{O}(h)$ and $\mathcal{O}\left(h^2\right)$, respectively. Finally, some numerical results are provided to confirm the theoretical analysis.

Numerical simulation of time-periodic problems is a special area of research, since the time periodicity modifies the problem structure, and then it is desirable to use parallel methods to solve such problems. The classical parareal algorithm for time-periodic problems, which is parallel in time, solving an initial value coarse problem, called the periodic parareal algorithm with initial value coarse problem (PP-IC), usually converges very slowly, and even diverges for wave propagation problems. In this paper, we first present a new PP-IC algorithm based on a diagonalization technique proposed recently. In this new algorithm, we approximate the coarse propagator G in the classical PP-IC algorithm with a head-tail coupled condition such that G can be parallelized using diagonalization in time. We analyze the convergence factors of the diagonalization-based PP-IC algorithm for both the linear and nonlinear cases. Then, we further design and analyze a new parallel-intime algorithm for time-periodic problems by combining the Krylov subspace method with the diagonalization-based PP-IC algorithm to accelerate the convergence. Finally, we also determine an appropriate choice of the parameter α in the head-tail coupling condition, and illustrate our theoretical results with several numerical experiments, both for model problems and the realistic application of Maxwell’s equations.

The weak convergence analysis plays an important role in error estimates for stochastic differential equations, which concerns with the approximation of the probability distribution of solutions. In this paper, we investigate the weak convergence order of a splitting-up method for stochastic differential equations. We first construct a splitting-up approximation, based on which we also set up a splitting-up numerical solution. We prove both of these two approximation methods are of first order of weak convergence with the help of Malliavin calculus. Finally, we present several numerical experiments to illustrate our theoretical analysis.

In this paper, a numerical method for solving nonlinear stochastic delay differential equations is proposed: two-step Milstein method. The mean square consistent and mean square convergence of the numerical method are studied. Through the relevant derivation, the conditions that the coefficients need to be satisfied when the numerical method is mean-square consistent and mean-square convergent are obtained, and it is proved that the mean-square convergence order of the numerical method is 1. Finally, the theoretical results are verified by numerical experiments.

In this paper, we present a novel Douglas-Rachford-splitting-based path following (DRS-PF) method that rapidly obtains the solution of linear programming (LP) with high accuracy. It originates from the fixed-point mapping associated with DRS on the log-barrier penalized LP. A path-following scheme is then proposed to simultaneously update the iterates and the penalty parameter for accelerating the overall procedure. Its global convergence towards an optimal solution to the original problem is established under mild assumptions. Numerical experiments show that DRS-PF outperforms the simplex and interior point methods implemented in the academic software (CLP, HiGHS, etc.) in terms of the geometric mean of the running time on a few typical benchmark data sets. In some cases, it is even reasonably competitive to the interior point method implemented in Gurobi, one of the most powerful software for LP.

In this work, we investigate the mechanism underlying loss spikes observed during neural network training. When the training enters a region with a lower-loss-as-sharper structure, the training becomes unstable, and the loss exponentially increases once the loss landscape is too sharp, resulting in the rapid ascent of the loss spike. The training stabilizes when it finds a flat region. From a frequency perspective, we explain the rapid descent in loss as being primarily influenced by low-frequency components. We observe a deviation in the first eigendirection, which can be reasonably explained by the frequency principle, as low-frequency information is captured rapidly, leading to the rapid descent. Inspired by our analysis of loss spikes, we revisit the link between the maximum eigenvalue of the loss Hessian (λ_max), flatness and generalization. We suggest that λ_max is a good measure of sharpness but not a good measure for generalization. Furthermore, we experimentally observe that loss spikes can facilitate condensation, causing input weights to evolve towards the same direction. And our experiments show that there is a correlation (similar trend) between λ_max and condensation. This observation may provide valuable insights for further theoretical research on the relationship between loss spikes, λ_max, and generalization.

For complex-valued or quaternionic neural networks, scholars and researchers usually decompose them into real-valued systems. The decomposed real-valued systems are equivalent to original systems. Then, the dynamical behaviors of real-valued systems obtained are investigated, including stability, synchronization, and chaos etc. In this paper, a class of quaternionic neural networks with time-varying delays is investigated. First, by designing a suitable PI controller, synchronization of the considered chaotic system is realized. By using a non-decomposition method and structuring a novel Lyapunov functional, sufficient conditions are derived to guarantee synchronization between the drive-response systems. It is worth mentioning that, unlike other methods, our approach does not require breaking down the quaternionic neural networks into four separate real-valued systems. Furthermore, we demonstrate the practical application of these chaotic quaternionic neural networks with time-varying delays in image encryption and decryption. Based on one sequence of chaotic signal from state trajectory of single quaternion-valued neuron and a new encryption algorithm, the application of chaotic system proposed, that is, image encryption, is researched. The process of image decryption is simply the reverse of the encryption process. Finally, numerical simulation examples are provided to validate the effectiveness of the designed PI controller and performance of image encryption and decryption.

In this paper, we develop a fully discrete virtual element scheme based on the local pressure projection stabilization for a three-field poroelasticity problem with a storage coefficient c₀ ≥ 0. We not only provide the well-posedness of the proposed scheme by proving a weaker form of the discrete inf-sup condition, but also show optimal error estimates for all unknowns, whose generic constants are independent of the Lamé coefficient λ. Moreover, our proposed scheme avoids pressure oscillation and applies to general polygonal elements, including hanging-node elements. Finally, we numerically validate the good performance of our virtual element scheme.

Superconvergence of differential structure on discretized surfaces is studied in this paper. The newly introduced geometric supercloseness provides us with a fundamental tool to prove the superconvergence of gradient recovery on deviated surfaces. An algorithmic framework for gradient recovery without exact geometric information is introduced. Several numerical examples are documented to validate the theoretical results.

The $k$-th ($k=3,4,5$) order backward differential formula ($\mathrm{BDF} k$) is applied to develop the high order energy stable schemes for the molecular beam epitaxial model with slope selection. The numerical schemes are established by combining the convex splitting technique with the $k$-th order accurate Douglas-Dupont stabilization term in the form of $S \tau^{k-1} \Delta_h\left(\phi^n-\phi^{n-1}\right)$. With the help of the new constructed discrete gradient structure of the $k$-th order explicit extrapolation formula, the stabilized $\mathrm{BDF} k$ scheme is proved to preserve energy dissipation law at the discrete levels and unconditionally stable in the energy norm. By using the discrete orthogonal convolution kernels and the associated convolution embedding inequalities, the $L^2$ norm error estimate is established under a weak constraint of time-step size. Numerical simulations are presented to demonstrate the accuracy and efficiency of the proposed numerical schemes.

This report presents a series of implicit-explicit (IMEX) variable stepsize algorithms for natural convection equations. The presented method requires a minimally intrusive modification to an existing program, does not add to the computational complexity, and is conceptually simple. Here, IMEX means the nonlinear term is treated fully explicitly, while the remaining terms are treated implicitly. Due to the increasing demand for low memory solvers, the addition of time adaptive can improve the accuracy and efficiency of the algorithms. For the first-order algorithm, we prove the stability of the variable stepsize backward Euler scheme combined with Adams-Bashforth 2 (VSS BE-AB2) and analyze convergence. Then, the stability of Constant Timestep Filtered-BE-AB2 (BE-AB2+F) is proved. Moreover, we construct adaptive algorithms by extending the approach to variable stepsize. Finally, numerical tests confirm the convergence rates of our method and validate the theoretical results.

In this paper, we propose a novel algorithm called neuron-wise parallel subspace correction method for the finite neuron method that approximates numerical solutions of partial differential equations (PDEs) using neural network functions. Despite extremely extensive research activities in applying neural networks for numerical PDEs, there is still a serious lack of effective training algorithms that can achieve adequate accuracy, even for one-dimensional problems. Based on recent results on the spectral properties of linear layers and analysis for single neuron problems, we develop a special type of subspace correction method that optimizes the linear layer and each neuron in the nonlinear layer separately. An optimal preconditioner that resolves the ill-conditioning of the linear layer is presented for one-dimensional problems, so that the linear layer is trained in a uniform number of iterations with respect to the number of neurons. In each single neuron problem, a local minimum is found by a superlinearly convergent algorithm. Numerical experiments on function approximation problems and PDEs demonstrate better performance of the proposed method than other gradient-based methods.

In this paper, we present a posteriori error estimates of the weak Galerkin finite element method for the steady-state Poisson-Nernst-Planck equations. The a posteriori error estimators for the electrostatic potential and ion concentrations are constructed. The reliability and efficiency of the estimators are verified by the upper and lower bounds of the energy norm of the error. The a posteriori error estimators are applied to the adaptive weak Galerkin algorithm for triangle, quadrilateral and polygonal meshes with hanging nodes. Finally, numerical results demonstrate the effectiveness of the adaptive algorithm guided by our constructed estimators.

The rigorous error analysis of a class of serendipity virtual element methods applied to numerically solve semilinear parabolic integro-differential equations on curved domains is the focus of this study. Different from the standard virtual element method, the serendipity virtual element method eliminates all the internal-moment degrees of freedom only under certain conditions of the mesh and the degree of approximation. Consequently, if the interpolation operators are utilized to approximate the nonlinear terms, the implementation of Newton’s iteration algorithm can be simplified. Nonhomogeneous Dirichlet boundary conditions are considered in this paper. The strategy of approximating curved domains with polygonal domains is taken into consideration, and to overcome the issue of suboptimal convergence caused by enforcing Dirichlet boundary conditions strongly, Nitsche-based projection method is employed to impose the boundary conditions weakly. For time discretization, Crank-Nicolson scheme incorporating trapezoidal quadrature rule is adopted. Based on the concrete formulation of Nitsche-based projection method, a Ritz-Volterra projection is introduced and its approximation properties are rigorously analyzed. Building upon these approximation properties, error estimates are derived for the fully discrete scheme. Additionally, the extension of the fully discrete scheme to 3D case is also included. Finally, we present two numerical experiments to corroborate the theoretical findings.