• Abstract

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.

• General Summary

• Abstract

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.

• General Summary

• Abstract

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 $\boldsymbol{A}$ 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 $\boldsymbol{A},$ 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 $ψ_h.$

• General Summary

• Abstract

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 $c_0 ≥ 0.$ 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.

• General Summary

• Abstract

This paper constructs the first mixed finite element for the linear elasticity problem in 3D using $P_3$ polynomials for the stress and discontinuous $P_2$ 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.

• General Summary

• Abstract

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 $r_k := τ_k/τ_{k−1} < r_{max} = 4.8645$ $(k ≥ 3)$ 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 $\mathcal{O}(N^{− min\{2,γα\}})$ is achieved, where $N$ denotes the total number of mesh points and $α$ indicates the regularity of the exact solution. This is, the optimal convergence will be achieved by taking $\gamma_{{\rm opt}} = 2/α.$ Numerical examples are provided to demonstrate our theoretical analysis.

• General Summary

• Abstract

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.

• General Summary

• Abstract

The aim of this paper is to present an improvement of the incremental Newton method proposed by Iannazzo [SIAM J. Matrix Anal. Appl., 28:2 (2006), 503–523] for approximating the principal $p$-th root of a matrix. We construct and analyze an incremental Chebyshev method with better numerical behavior. We present a convergence and numerical analysis of the method, where we compare it with the corresponding incremental Newton method. The new method has order of convergence three and is stable and more efficient than the incremental Newton method.

• General Summary

• Abstract

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.

• General Summary

• Abstract

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 $(\phi, \mathbf{θ})$ and the bilinear Lagrange approximation for temperature $u.$ In terms of the special properties of these elements above, the superclose error estimates with order $\mathcal{O}(h^2)$ are obtained firstly for all three components in such a strongly coupled system. Subsequently, the global superconvergence error estimates with order $\mathcal{O}(h^2)$ 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 $\mathcal{O}(h^2 ),$ respectively. Finally, some numerical results are provided to confirm the theoretical analysis.

• General Summary

• Abstract

We study decentralized smooth optimization problems over compact submanifolds. Recasting it as a composite optimization problem, we propose a decentralized Douglas-Rachford 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.

• General Summary

• Abstract

In a previous paper, a technique was suggested to avoid order reduction with any explicit exponential Runge-Kutta method when integrating initial boundary value nonlinear problems with time-dependent boundary conditions. In this paper, we significantly simplify the full discretization formulas to be applied under conditions which are nearly always satisfied in practice. Not only a simpler linear combination of $\varphi_j$-functions is given for both the stages and the solution, but also the information required on the boundary is so much simplified that, in order to get local order three, it is no longer necessary to resort to numerical differentiation in space. In many cases, even to get local order 4. The technique is then shown to be computationally competitive against other widely used methods with high enough stiff order through the standard method of lines.

• General Summary

• Abstract

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.

• General Summary

• Abstract

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 $L^2$ 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.

• General Summary

• Abstract

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.

• General Summary

• Abstract

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.

• General Summary

• Abstract

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 eigenfunction 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.

• General Summary

• Abstract

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.

• General Summary

• Abstract

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 $Ω = [a, b].$ 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 $x = ξ ∈ (a, b),$ 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 $[a, ξ]$ and $[ξ, b],$ which ensures that $x = ξ$ 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.

• General Summary

• Abstract

The $k$-th ($k = 3, 4, 5$) order backward differential formula (${\rm BDF}k$) 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 $S\tau^{k−1}∆_h(\phi^n−\phi^{n−1}).$ With the help of the new constructed discrete gradient structure of the $k$-th order explicit extrapolation formula, the stabilized ${\rm BDF}k$ 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.

• General Summary