We develop a stabilizer free weak Galerkin (SFWG) finite element method for Brinkman equations. The main idea is to use high order polynomials to compute the discrete weak gradient and then the stabilizing term is removed from the numerical formulation. The SFWG scheme is very simple and easy to implement on polygonal meshes. We prove the well-posedness of the scheme and derive optimal order error estimates in energy and $L^2$ norm. The error results are independent of the permeability tensor, hence the SFWG method is stable and accurate for both the Stokes and Darcy dominated problems. Finally, we present some numerical experiments to verify the efficiency and stability of the SFWG method.

The aim of this paper is to develop a fast multigrid solver for interpolation-free finite volume (FV) discretization of anisotropic elliptic interface problems on general bounded domains that can be described as a union of blocks. We assume that the curved interface falls exactly on the boundaries of blocks. The transfinite interpolation technique is applied to generate block-wise distorted quadrilateral meshes, which can resolve the interface with fine geometric details. By an extensive study of the harmonic average point method, an interpolation-free nine-point FV scheme is then derived on such multi-block grids for anisotropic elliptic interface problems with non-homogeneous jump conditions. Moreover, for the resulting linear algebraic systems from cell-centered FV discretization, a high-order prolongation operator based fast cascadic multigrid solver is developed and shown to be robust with respect to both the problem size and the jump of the diffusion coefficients. Various non-trivial examples including four interface problems and an elliptic problem in complex domain without interface, all with tens of millions of unknowns, are provided to show that the proposed multigrid solver is dozens of times faster than the classical algebraic multigrid method as implemented in the code AMG1R5 by Stüben.

Given the measurement matrix $A$ and the observation signal $y,$ the central purpose of compressed sensing is to find the most sparse solution of the underdetermined linear system $y = Ax+z,$ where $x$ is the $s$-sparse signal to be recovered and $z$ is the noise vector. Zhou and Yu [Front. Appl. Math. Stat., 5 (2019), Article 14] recently proposed a novel non-convex weighted $ℓ_r−ℓ_1$ minimization method for effective sparse recovery. In this paper, under newly coherence-based conditions, we study the non-convex weighted $ℓ_r −ℓ_1$ minimization in reconstructing sparse signals that are contaminated by different noises. Concretely, the results reveal that if the coherence $\mu$ of measurement matrix $A$ fulfills $$\mu < \kappa (s; r, α, N), s > 1, α^{\frac{1}{r}} N^{\frac{1}{2}} < 1,$$ then any $s$-sparse signals in the noisy scenarios could be ensured to be reconstructed robustly by solving weighted $ℓ_r − ℓ_1$ minimization non-convex optimization problem. Furthermore, some central remarks are presented to clear that the reconstruction assurance is much weaker than the existing ones. To the best of our knowledge, this is the first mutual coherence-based sufficient condition for such approach.

In this paper, we investigate the local discontinuous Galerkin method with generalized numerical fluxes for one-dimensional nonlinear Korteweg-de Vries type equations. The numerical flux for the nonlinear convection term is chosen as the generalized Lax-Friedrichs flux, and the generalized alternating flux and upwind-biased flux are used for the dispersion term. The generalized Lax-Friedrichs flux with anti-dissipation property will compensate the numerical dissipation of the dispersion term, resulting in a nearly energy conservative scheme that is useful in resolving waves and is beneficial for long time simulations. To deal with the nonlinearity and different numerical flux weights, a suitable numerical initial condition is constructed, for which a modified global projection is designed. By establishing relationships between the prime variable and auxiliary variables in combination with sharp bounds for jump terms, optimal error estimates are obtained. Numerical experiments are shown to confirm the validity of theoretical results.

This paper presents space-time continuous and time discontinuous Galerkin schemes for solving nonlinear time-fractional partial differential equations based on B-splines in time and non-uniform rational B-splines (NURBS) in space within the framework of Isogeometric Analysis. The first approach uses the space-time continuous Petrov-Galerkin technique for a class of nonlinear time-fractional Sobolev-type equations and the optimal error estimates are obtained through a concise equivalence analysis. The second approach employs a generalizable time discontinuous Galerkin scheme for the time-fractional Allen-Cahn equation. It first transforms the equation into a time integral equation and then uses the discontinuous Galerkin method in time and the NURBS discretization in space. The optimal error estimates are provided for the approach. The convergence analysis under time graded meshes is also carried out, taking into account the initial singularity of the solution for two models. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed methods.

In this paper, we propose two families of nonconforming finite elements on $n$-rectangle meshes of any dimension to solve the sixth-order elliptic equations. The unisolvent property and the approximation ability of the new finite element spaces are established. A new mechanism, called the exchange of sub-rectangles, for investigating the weak continuities of the proposed elements is discovered. With the help of some conforming relatives for the $H^3$ problems, we establish the quasi-optimal error estimate for the triharmonic equation in the broken $H^3$ norm of any dimension. The theoretical results are validated further by the numerical tests in both 2D and 3D situations.

This paper presents a stochastic modification of a limited memory BFGS method to solve bound-constrained global minimization problems with a differentiable cost function with no further smoothness. The approach is a stochastic descent method where the deterministic sequence, generated by a limited memory BFGS method, is replaced by a sequence of random variables. To enhance the performance of the proposed algorithm and make sure the perturbations lie within the feasible domain, we have developed a novel perturbation technique based on truncating a multivariate double exponential distribution to deal with bound-constrained problems; the theoretical study and the simulation of the developed truncated distribution are also presented. Theoretical results ensure that the proposed method converges almost surely to the global minimum. The performance of the algorithm is demonstrated through numerical experiments on some typical test functions as well as on some further engineering problems. The numerical comparisons with stochastic and meta-heuristic methods indicate that the suggested algorithm is promising.

In this paper, a robust residual-based a posteriori estimate is discussed for the Streamline Upwind/Petrov Galerkin (SUPG) virtual element method (VEM) discretization of convection dominated diffusion equation. A global upper bound and a local lower bound for the a posteriori error estimates are derived in the natural SUPG norm, where the global upper estimate relies on some hypotheses about the interpolation errors and SUPG virtual element discretization errors. Based on the Dörfler’s marking strategy, adaptive VEM algorithm drived by the error estimators is used to solve the problem on general polygonal meshes. Numerical experiments show the robustness of the a posteriori error estimates.

This article presents an image space branch-reduction-bound algorithm for globally solving the sum of affine ratios problem. The algorithm works by solving its equivalent problem, and by using convex hull and concave hull approximation of bilinear function, we can construct the affine relaxation problem of the equivalent problem, which can be used to compute the lower bounds during the branch-and-bound search. By subsequently refining the initial image space rectangle and solving a series of affine relaxation problems, the proposed algorithm is convergent to the global optima of the primal problem. For improving the convergence speed, an image space region reducing method is adopted for compressing the investigated image space rectangle. In addition, the global convergence of the algorithm is proved, and its computational complexity is analyzed. Finally, comparing with some existing methods, numerical results indicate that the algorithm has better computational performance.

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.