A special finite volume element method based on postprocessing technique is proposed to solve the anisotropic diffusion problem on arbitrary convex polygonal meshes. The shape function of polygonal finite element method is constructed by Wachspress generalized barycentric coordinate, and by adding some element-wise bubble functions to the finite element solution, we get a new finite volume element solution that satisfies the local conservation law on a certain dual mesh. The postprocessing algorithm only needs to solve a local linear algebraic system on each primary cell, so that it is easy to implement. More interesting is that, a general construction of the bubble functions is introduced on each polygonal cell, which enables us to prove the existence and uniqueness of the post-processed solution on arbitrary convex polygonal meshes with full anisotropic diffusion tensor. The optimal $H^1$ and $L^2$ error estimates of the post-processed solution are also obtained. Finally, the local conservation property and convergence of the new polygonal finite volume element solution are verified by numerical experiments.

In this paper, we present and analyze a posteriori error estimates in the $L^2$-norm of an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order boundary-value problems for ordinary differential equations of the form $u′′=f(x, u).$ We first use the superconvergence results proved in the first part of this paper (J. Appl. Math. Comput. 69, 1507-1539, 2023) to prove that the UWDG solution converges, in the $L^2$-norm, towards a special $p$-degree interpolating polynomial, when piecewise polynomials of degree at most $p ≥ 2$ are used. The order of convergence is proved to be $p + 2.$ We then show that the UWDG error on each element can be divided into two parts. The dominant part is proportional to a special $(p+1)$-degree Baccouch polynomial, which can be written as a linear combination of Legendre polynomials of degrees $p − 1,$ $p,$ and $p + 1.$ The second part converges to zero with order $p + 2$ in the $L^2$-norm. These results allow us to construct a posteriori UWDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the $L^2$-norm under mesh refinement. The order of convergence is proved to be $p + 2.$ Finally, we prove that the global effectivity index converges to unity at $\mathcal{O}(h)$ rate. Numerical results are presented exhibiting the reliability and the efficiency of the proposed error estimator.

A new weak Galerkin method with weakly enforced Dirichlet boundary condition is proposed and analyzed for the second order elliptic problems. Two penalty terms are incorporated into the weak Galerkin method to enforce the boundary condition in the weak sense. The new numerical scheme is designed by using the locally constructed weak gradient. Optimal order error estimates are established for the numerical approximation in the energy norm and usual $L^2$ norm. Moreover, by using the Schur complement technique, the unknowns of the numerical scheme are only defined on the boundary of each piecewise element and an effective implementation of the reduced global system is presented. Some numerical experiments are reported to demonstrate the accuracy and efficiency of the proposed method.

In this paper we develop convergence and acceleration theory for Anderson acceleration applied to Newton’s method for nonlinear systems in which the Jacobian is singular at a solution. For these problems, the standard Newton algorithm converges linearly in a region about the solution; and, it has been previously observed that Anderson acceleration can substantially improve convergence without additional a priori knowledge, and with little additional computation cost. We present an analysis of the Newton-Anderson algorithm in this context, and introduce a novel and theoretically supported safeguarding strategy. The convergence results are demonstrated with the Chandrasekhar H-equation and a variety of benchmark examples.

The physics-informed neural networks (PINNs) has been widely utilized to numerically approximate PDE problems. While PINNs has achieved good results in producing solutions for many partial differential equations, studies have shown that it does not perform well on phase field models. In this paper, we partially address this issue by introducing a modified physics-informed neural networks. In particular, they are used to numerically approximate Allen-Cahn-Ohta-Kawasaki (ACOK) equation with a volume constraint. Technically, the inverse of Laplacian in the ACOK model presents many challenges to the baseline PINNs. To take the zero-mean condition of the inverse of Laplacian, as well as the volume constraint, into consideration, we also introduce a separate neural network, which takes the second set of sampling points in the approximation process. Numerical results are shown to demonstrate the effectiveness of the modified PINNs. An additional benefit of this research is that the modified PINNs can also be applied to learn more general nonlocal phase-field models, even with an unknown nonlocal kernel.

This paper focuses on neural network-based learning methods for identifying nonlinear dynamic systems. The Takagi-Sugeno (T-S) fuzzy model is introduced to represent nonlinear systems in a linear way. Fractional calculus is integrated to minimize the cost function, yielding a fractional-order learning algorithm that can derive optimal parameters in the T-S fuzzy model. The proposed algorithm is evaluated by comparing it with an integer-order method for identifying numerical nonlinear systems and a water quality system. Both evaluations demonstrate that the proposed algorithm can effectively reduce errors and improve model accuracy.

In this paper, we develop a novel meshless, ray-based deep neural network algorithm for solving the high-frequency Helmholtz scattering problem in the unbounded domain. While our recent work [44] designed a deep neural network method for solving the Helmholtz equation over finite bounded domains, this paper deals with the more general and difficult case of unbounded regions. By using the perfectly matched layer method, the original mathematical model in the unbounded domain is transformed into a new format of second-order system in a finite bounded domain with simple homogeneous Dirichlet boundary conditions. Compared with the Helmholtz equation in the bounded domain, the new system is equipped with variable coefficients. Then, a deep neural network algorithm is designed for the new system, where the rays in various random directions are used as the basis of the numerical solution. Various numerical examples have been carried out to demonstrate the accuracy and efficiency of the proposed numerical method. The proposed method has the advantage of easy implementation and meshless while maintaining high accuracy. To the best of the author’s knowledge, this is the first deep neural network method to solve the Helmholtz equation in the unbounded domain.