Extended finite element method with edge-based strain smoothing (ESm-XFEM) for linear elastic crack growth

In this paper, we use the mixed direct discontinuous Galerkin method (DDG) to solve the biharmonic equation. Firstly, by introducing an auxiliary variable, the biharmonic equation is split into two second-order equations. Secondly, the variational problem based on the DDG method of the system is derived and its well-posedness is proven. Next, error estimates of the approximate solution in L 2 norm and energy norm are present. For a given polynomial degree k (k ≥ 1), the optimal convergence rates concerning energy norm and norm are k and k + 1, respectively. Finally, numerical results demonstrate the accuracy and capability of the method.

We consider the wavelet-based estimators of mean regression function with long memory moving average errors and investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence rates within a logarithmic term over a large range of Besov function classes \(B^{s}_{p,q}\). Therefore, in the presence of long memory non-Gaussian moving average noise, wavelet estimators still achieve nearly optimal convergence rates and provide explicitly the extraordinary local adaptability. The theory is illustrated with some numerical examples.

Conforming virtual element approximations of the two-dimensional Stokes problem

A numerical method for a two-dimensional curl–curl and grad-div problem is studied in this paper. It is based on a discretization using weakly continuous P 1 vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive $${\epsilon}$$) in both the energy norm and the L 2 norm are established on graded meshes. The theoretical results are confirmed by numerical experiments.

We propose and analyze an unfitted finite element method of arbitrary order for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. We apply a non-symmetric bilinear form and the boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. The optimal convergence rate under the energy norm is derived, and \(O(h^{-2})\)-upper bounds of the condition numbers are shown for the final linear systems. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.

A discontinuous Galerkin least-squares method for div–curl systems

In this paper, Newmark time-stepping scheme and edge elements are used to numerically solve the time-dependent scattering problem in a three-dimensional cavity. Finite element methods based on the variational formulation derived in [23] are considered. Due to the lack of regularity ofer, the existence and uniqueness of the discrete solutions and their convergence are proved by using the concept of collectively compact operators. An optimal convergence rate in the energy norm is also established.

Stable GFEM (SGFEM): Improved conditioning and accuracy of GFEM/XFEM for three-dimensional fracture mechanics

Models built with deep neural network (DNN) can handle complicated real-world data extremely well, seemingly without suffering from the curse of dimensionality or the non-convex optimization. To contribute to the theoretical understanding of deep learning, this work studies the nonparametric perspective of DNNs by considering the following questions: (1) What is the underlying estimation problem and what are the most appropriate data assumptions? (2) What is the corresponding optimal convergence rate and does the curse of dimensionality occur? (3) Is the optimal rate achievable for DNN estimators and is there any optimization guarantee? These questions are investigated on two of the most fundamental problems --- regression and classification. Specifically, statistical optimality of DNN estimators is established under various settings with special focuses on the curse of dimensionality and optimization guarantee.In the classic binary classification problem, statistical optimal convergence rates that suffer less from the curse of dimensionality are established under two settings:(1) Under the smooth boundary assumption, I show that DNN classifiers with proper architectures can benefit from the compositional smoothness structure underlying the high dimensional data in the sense that the optimal convergence rates only depend on some effective dimension d*, potentially much smaller than the data dimension d. (2) Under a novel teacher-student framework that assumes the Bayes classifier to be expressed as ReLU neural networks, I obtain a dimension-free rate of convergence O(n^{-2/3}) for DNN classifiers, which is also proven optimal. The optimization of DNN is highly complicated with generally no algorithmic guarantee on finding the global minimizer. To this end, I turn to the recently proposed neural tangent kernel (NTK) literature where the similarity between overparametrized DNN trained by gradient descent (GD) and kernel methods are established. Specifically, through a comprehensive analysis of L2-regularized GD trajectories, I prove that for overparametrized one-hidden-layer ReLU neural networks with L2 regularization, the output from GD is close to that from the kernel ridge regression with the corresponding NTK and optimal rate of L2 estimation error can be achieved.

Locally discontinuous but globally continuous Galerkin methods for elliptic problems

Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach

A hierarchical approach to the a posteriori error estimation of isogeometric Kirchhoff plates and Kirchhoff–Love shells

Simulation of thermoelastic crack problems using singular edge-based smoothed finite element method

Hypersonic similarity for steady compressible full Euler flows over two-dimensional Lipschitz wedges

Kirchhoff–Love shell formulation based on triangular isogeometric analysis