Sequential Approximation of Functions in Sobolev Spaces Using Random Samples Abstract We present an iterative algorithm for approximating an unknown function sequentially using random samples of the function values and gradients. This is an extension of the recently developed sequential approximation (SA) method, which approximates a target function using samples of function values only. The current paper extends the development of the SA methods to the Sobolev space and allows the use of gradient information naturally. The algorithm is easy to implement, as it requires only vector operations and does not involve any matrices. We present tight error bound of the algorithm, and derive an optimal sampling probability measure that results in fastest error convergence. Numerical examples are provided to verify the theoretical error analysis and the effectiveness of the proposed SA algorithm.

An Adaptive hp –DG–FE Method for Elliptic Problems: Convergence and Optimality in the 1D Case Abstract We propose and analyze an hp-adaptive DG–FEM algorithm, termed \(\varvec {hp}\) -ADFEM, and its one-dimensional realization, which is convergent, instance optimal, and h- and p-robust. The procedure consists of iterating two routines: one hinges on Binev’s algorithm for the adaptive hp-approximation of a given function, and finds a near-best hp-approximation of the current discrete solution and data to a desired accuracy; the other one improves the discrete solution to a finer but comparable accuracy, by iteratively applying Dörfler marking and h refinement.

A Review on Stochastic Multi-symplectic Methods for Stochastic Maxwell Equations Abstract Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al. (A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255–268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.

A Note on the Adaptive Simpler Block GMRES Method Abstract The adaptive simpler block GMRES method was investigated by Zhong et al. (J Comput Appl Math 282:139–156, 2015) where the condition number of the adaptively chosen basis for the Krylov subspace was evaluated. In this paper, the new upper bound for the condition number is investigated. Numerical tests show that the new upper bound is tighter.

$$C^1$$ C 1 -Conforming Quadrilateral Spectral Element Method for Fourth-Order Equations Abstract This paper is devoted to Professor Benyu Guo’s open question on the \(C^1\) -conforming quadrilateral spectral element method for fourth-order equations which has been endeavored for years. Starting with generalized Jacobi polynomials on the reference square, we construct the \(C^1\) -conforming basis functions using the bilinear mapping from the reference square onto each quadrilateral element which fall into three categories—interior modes, edge modes, and vertex modes. In contrast to the triangular element, compulsively compensatory requirements on the global \(C^1\) -continuity should be imposed for edge and vertex mode basis functions such that their normal derivatives on each common edge are reduced from rational functions to polynomials, which depend on only parameters of the common edge. It is amazing that the \(C^1\) -conforming basis functions on each quadrilateral element contain polynomials in primitive variables, the completeness is then guaranteed and further confirmed by the numerical results on the Petrov–Galerkin spectral method for the non-homogeneous boundary value problem of fourth-order equations on an arbitrary quadrilateral. Finally, a \(C^1\) -conforming quadrilateral spectral element method is proposed for the biharmonic eigenvalue problem, and numerical experiments demonstrate the effectiveness and efficiency of our spectral element method.

A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials Abstract In this work, we apply a semi-Lagrangian spectral method for the Vlasov–Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space–velocity domain with a BDF time-stepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on some standard benchmark problems including the two-stream instability and the Landau damping test cases. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure.

The INTERNODES Method for Non-conforming Discretizations of PDEs Abstract INTERNODES is a general purpose method to deal with non-conforming discretizations of partial differential equations on 2D and 3D regions partitioned into two or several disjoint subdomains. It exploits two intergrid interpolation operators, one for transfering the Dirichlet trace across the interfaces, and the other for the Neumann trace. In this paper, in every subdomain the original problem is discretized by either the finite element method (FEM) or the spectral element method (SEM or hp-FEM), using a priori non-matching grids and piecewise polynomials of different degrees. Other discretization methods, however, can be used. INTERNODES can also be applied to heterogeneous or multiphysics problems, that is, problems that feature different differential operators inside adjacent subdomains. For instance, in this paper we apply the INTERNODES method to a Stokes–Darcy coupled problem that models the filtration of fluids in porous media. Our results highlight the flexibility of the method as well as its optimal rate of convergence with respect to the grid size and the polynomial degree.

Multigrid Methods for Time-Fractional Evolution Equations: A Numerical Study Abstract In this work, we develop an efficient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of order \(\alpha (0,1)\) in time. The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear finite elements in space and corrected high-order BDF convolution quadrature in time. At each time step, instead of solving the linear algebraic system exactly, we employ a multigrid iteration with a Gauss–Seidel smoother to approximate the solution efficiently. Illustrative numerical results for nonsmooth problem data are presented to demonstrate the approach.

Finite Element Convergence for State-Based Peridynamic Fracture Models Abstract We establish the a priori convergence rate for finite element approximations of a class of nonlocal nonlinear fracture models. We consider state-based peridynamic models where the force at a material point is due to both the strain between two points and the change in volume inside the domain of the nonlocal interaction. The pairwise interactions between points are mediated by a bond potential of multi-well type while multi-point interactions are associated with the volume change mediated by a hydrostatic strain potential. The hydrostatic potential can either be a quadratic function, delivering a linear force–strain relation, or a multi-well type that can be associated with the material degradation and cavitation. We first show the well-posedness of the peridynamic formulation and that peridynamic evolutions exist in the Sobolev space \(H^2\) . We show that the finite element approximations converge to the \(H^2\) solutions uniformly as measured in the mean square norm. For linear continuous finite elements, the convergence rate is shown to be \(C_t \Delta t + C_s h^2/\epsilon ^2\) , where \(\epsilon \) is the size of the horizon, h is the mesh size, and \(\Delta t\) is the size of the time step. The constants \(C_t\) and \(C_s\) are independent of \(\Delta t\) and h and may depend on \(\epsilon \) through the norm of the exact solution. We demonstrate the stability of the semi-discrete approximation. The stability of the fully discrete approximation is shown for the linearized peridynamic force. We present numerical simulations with the dynamic crack propagation that support the theoretical convergence rate.

The Nonlinear Lopsided HSS-Like Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems with Positive-Definite Matrices Abstract In this paper, by means of constructing the linear complementarity problems into the corresponding absolute value equation, we raise an iteration method, called as the nonlinear lopsided HSS-like modulus-based matrix splitting iteration method, for solving the linear complementarity problems whose coefficient matrix in \(\mathbb {R}^{n\times n}\) is large sparse and positive definite. From the convergence analysis, it is appreciable to see that the proposed method will converge to its accurate solution under appropriate conditions. Numerical examples demonstrate that the presented method precede to other methods in practical implementation.