In this article, we propose the time discretization of the second order decoupled implicit/explicit method of the 3D primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side. We deduce the second order optimal error estimates on the $L^2$ and $H^1$ norms of the time discrete velocity and density and the $L^2$ norm of the time discrete pressure under the restriction of the time step $0<\tau\leq\beta$ for some positive constant $\beta$. Also, we deduce some stability results on the time discrete solution under the same restriction on the time step.

This paper introduces a new weak Galerkin (WG) finite element method for second order elliptic equations on polytopal meshes. This method, called WG-FEM, is designed by using a discrete weak gradient operator applied to discontinuous piecewise polynomials on finite element partitions of arbitrary polytopes with certain shape regularity. The paper explains how the numerical schemes are designed and why they provide reliable numerical approximations for the underlying partial differential equations. In particular, optimal order error estimates are established for the corresponding WG-FEM approximations in both a discrete $H^1$ norm and the standard $L^2$ norm. Numerical results are presented to demonstrate the robustness, reliability, and accuracy of the WG-FEM. All the results are established for finite element partitions with polytopes that are shape regular.

It is well-known that classical Dirichlet-Neumann loosely coupled partitioned schemes for fluid-structure interaction (FSI) problems are unconditionally unstable for certain combinations of physical and geometric parameters that are relevant in hemodynamics. It was shown in [18] on a simple test problem, that these instabilities are associated with the so called “added-mass effect”. By considering the same test problem as in [18], the present work shows that a novel, partitioned, loosely coupled scheme, recently introduced in [11], called the kinematically coupled β-scheme, does not suffer from the added mass effect for any β ∈ [0; 1], and is unconditionally stable for all the parameters in the problem. Numerical results showing unconditional stability are presented for a full, nonlinearly coupled benchmark FSI problem, first considered in [31].

The focal point of this paper is to propose and analyze a $\mathbb{P}_0$ discontinuous Galerkin (DG) formulation for image denoising. The scheme is based on a total variation approach which has been applied successfully in previous papers on image processing. The main idea of the new scheme is to model the restoration process in terms of a discrete energy minimization problem and to derive a corresponding DG variational formulation. Furthermore, we will prove that the method exhibits a unique solution and that a natural maximum principle holds. In addition, a number of examples illustrate the effectiveness of the method.

In this paper we define a finite difference method for the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamical sense perfect and polytropic. The homogeneous boundary conditions for velocity, microrotation and heat flux are proposed. The sequence of approximate solutions for our problem is constructed by using the defined finite difference approximate equations system. We investigate the properties of these approximate solutions and establish their convergence to the strong solution of our problem globally in time, which is the main results of the paper. A numerical experiment is performed by solving the defined approximate ordinary differential equations system using strong-stability preserving (SSP) Runge-Kutta scheme for time discretization.

We study the numerical approximation of a flow problem governed by a viscoelastic model coupled with a one-dimensional elastic structure equation. A variational formulation for flow equations is developed based on the Arbitrary Lagrangian-Eulerian (ALE) method and stability of the time discretized system is investigated. For decoupled numerical algorithms we consider both an explicit scheme of the Leap-Frog type and a fully implicit scheme.

In this paper, we develop the second-order two-scale (SOTS) analysis method and numerical algorithm for dynamic thermo-mechanical problems of composite materials with 3-D periodic configuration. In the problem considered, there exists a mutual interaction between the displacement and temperature fields. By the asymptotic expansion of temperature and displacement fields, the cell problems, effective thermal and mechanical parameters, homogenized equations and SOTS formulas of temperatures and displacements are obtained. The numerical algorithm based on the SOTS method is given. Finally, some numerical examples are shown. The numerical results show that the SOTS method is feasible and valid to predict the dynamic thermo-mechanical behaviors of periodic composite materials.

We present and analyze a posteriori error estimates for the local discontinuous Galerkin (LDG) method for the linearized Korteweg-de Vries (KdV) equation in one space dimension. These estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We extend the work of Hufford and Xing [J. Comput. Appl. Math., 255 (2014), pp. 441-455] to prove new superconvergence results towards particular projections of the exact solutions for the two auxiliary variables in the LDG method that approximate the first and second derivatives of the solution. The order of convergence is proved to be $k+3/2$, when polynomials of total degree not exceeding $k$ are used. These results allow us to prove that the significant parts of the spatial discretization errors for the LDG solution and its spatial derivatives (up to second order) are proportional to $(k+1)$-degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates and prove that, for smooth solutions, these a $posteriori$ LDG error estimates for the solution and its spatial derivatives, at a fixed time $t$, converge to the true errors at $\mathcal{O}(h^{k+3/2})$ rate in the $L^2$-norm. Finally, we prove that the global effectivity indices, for the solution and its spatial derivatives, converge to unity at $\mathcal{O}(h^{\frac{1}{2}})$ rate. Numerical results are presented to validate the theory.