Ran Zhang's research while affiliated with Jilin University and other places

Publications (16)

Preprint
In this paper, we develop a new weak Galerkin finite element scheme for the Stokes interface problem with curved interfaces. We take a unique vector-valued function at the interface and reflect the interface condition in the variational problem. Theoretical analysis and numerical experiments show that the errors can reach the optimal convergence or...
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In this paper, we study nonlinear filtering problems via solving their corresponding Zakai equations. Using the splitting-up technique, we approximate the Zakai equation with two equations consisting of a first-order stochastic partial differential equation and a deterministic second-order partial differential equation. For the splitting-up equatio...
Preprint
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The present paper proposes a new framework for describing the stock price dynamics. In the traditional geometric Brownian motion model and its variants, volatility plays a vital role. The modern studies of asset pricing expand around volatility, trying to improve the understanding of it and remove the gap between the theory and market data. Unlike...
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Based on the auxiliary subspace techniques, a hierarchical basis a posteriori error estimator is proposed for the Stokes problem in two and three dimensions. For the error estimator, we need to solve only two global diagonal linear systems corresponding to the degree of freedom of velocity and pressure respectively, which reduces the computational...
Preprint
Based on the auxiliary subspace techniques, a hierarchical basis a posteriori error estimator is proposed for the Stokes problem in two and three dimensions. For the error estimator, we need to solve only two global diagonal linear systems corresponding to the degree of freedom of velocity and pressure respectively, which reduces the computational...
Article
In this paper, we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition. We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation. A discrete inf-sup condition is proved and the optimal error estimates are al...
Preprint
In this paper, we propose a new numerical scheme for the coupled Stokes-Darcy model with Beavers-Joseph-Saffman interface condition. We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to the Darcy equation. A discrete inf-sup condition is proved and optimal error estimates are also derived. Numeric...
Article
This article presents a high order conservative flux optimization (CFO) finite element method that preserve the important mass conservation property locally for the convection-diffusion equations. The numerical scheme is an extension of the first order CFO scheme proposed in [25], it is based on the classical Galerkin finite element method enhanced...
Article
We reformulate the second order non-divergence form elliptic equations in a symmetric variation form, i.e., a least-squares form. We design a weak Galerkin finite element method for this high-regularity formulation. The optimal order of convergence is proved. Numerical results verify the theory. In addition, numerical results, compared to the exist...
Preprint
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In this paper, we propose a dynamical model to describe the transmission of COVID-19, which is spreading in China and many other countries. To avoid a larger outbreak in the worldwide, Chinese government carried out a series of strong strategies to prevent the situation from deteriorating. Home quarantine is the most important one to prevent the sp...
Article
In this paper, we study the existence, uniqueness, and regularity of solutions for the nonlinear functional equations y(t) =b(t,y(θ(t)))+f(t),t∈[0,T], with vanishing delay θ(t). We then present a collocation method to solve this equation, and analyze the convergence properties of the piecewise polynomial collocation approximations. Several numerica...
Preprint
This article presents a high order conservative flux optimization (CFO) finite element method for the elliptic diffusion equations. The numerical scheme is based on the classical Galerkin finite element method enhanced by a flux approximation on the boundary of a prescribed set of arbitrary control volumes (either the finite element partition itsel...
Article
Full-text available
Recently, we proposed a weak Galerkin finite element method for the Laplace eigenvalue problem. In this paper, we present two-grid and two-space skills to accelerate the weak Galerkin method. By choosing parameters properly, the two-grid and two-space weak Galerkin method not only doubles the convergence rate, but also maintains the asymptotic lowe...
Article
In this paper, we investigate the weak Galerkin method for the Laplacian eigenvalue problem. We use the weak Galerkin method to obtain lower bounds of Laplacian eigenvalues, and apply a postprocessing technique to get upper bounds. Thus, we can verify the accurate intervals which the exact eigenvalues lie in. This postprocessing technique is effcie...
Article
This article presents a weak Galerkin (WG) finite element method for the Cahn-Hilliard equation. The WG method makes use of piecewise polynomials as approximating functions, with weakly defined partial derivatives (first and second order) computed locally by using the information in the interior and on the boundary of each element. A stabilizer is...

Citations

... Later, the WG method was also used to solve the Stokes equation [22,25], Brinkman equation [10,23], linear elasticity equation [20]. Due to the discontinuity of weak functions, the WG is more flexible in dealing with interface problems [9,11,13]. For example, when dealing with elliptic interface problems, it can be the same value on the interface [11], and the interface condition is reflected in the variational problem, or it can be two values, and the interface condition is transformed into Dirichlet boundary condition. ...
... An initial condition u 0 and a boundary condition g(x) on ∂ for u are prescribed. a = a(x, t) and b = b(x, t) are the diffusion and convection coefficients; the right-hand side, f = f (x, t) represents the source function [38]. To solve the above model with the finite element method, C-Bézier and H-Bézier basis functions are selected to construct trial and test function spaces. ...
... Keywords Eigenvalue problems · Upper and lower bound · Weak Galerkin method · High accuracy 1 Introduction Wang et al. [11,12] first introduced a weak Galerkin method for the second order elliptic problems, which was further extended to other model problems like Stokes equations [13], Maxwell equations [9], etc. The WG method is considered for eigenvalue problems (EVP for short hereinafter) in [6,17], where lower bounds of eigenvalues are derived for elliptic eigenvalue problems. Actually, the monotonicity of approximate eigenvalues is a quite attractive topic. ...
... Armentano and Durán in [27] first used the CR element to get the asymptotic lower bounds for the eigenvalues of the Laplace operator. On the basis of their work, finding the asymptotic lower bounds of eigenvalues was further developed by many researchers (e.g., see [28][29][30][31][32][33][34][35][36][37][38]). In recent years, the guaranteed lower bounds for eigenvalues have attracted academic attention. ...
... More details of this method can be found in Lin et al. [2018], Mu et al. [2015], Wang and Ye [2014] and references therein. In recent times, WG finite element method is used for different models, like the Cahn-Hilliard model is solved in Wang et al. [2019a] and Biot's consolidation model is analyzed in Hu et al. [2018]. In this work, we will use a modified version of the weak Galerkin method introduced in Wang et al. [2014], Gao et al. [2020] for spatial discretization. ...
... In modern scientific and engineering applications, the eigenvalue problems arising from partial differential equations(PDEs) are of fundamental importance in many fields [1][2][3][4]. There are many numerical methods to solve the Laplace eigenvalue problem [5][6][7][8] and finite element method (FEM) is an effective numerical method for the eigenvalue problem [9][10][11][12][13]. The virtual element method (VEM), introduced in [14], is a successful extension of FEM to polygonal/polyhedral meshes. ...