In this paper, a generalized particle system (GPS) method, a general method to describe multiple strong form representation based particle methods is described. Gradient, divergence, and Laplacian operators used in various strong form based particle method such as moving particle semi-implicit (MPS) method, smooth particle hydrodynamics (SPH), and peridynamics, can be described by the GPS metho...

We derive explicit formulas for the eigenvalues and eigenvectors of the Discrete Laplacian on a rectangular grid for the standard finite difference and finite element methods in 1D, 2D, and 3D. Periodic, Dirichlet, Neumann, and mixed boundary conditions are all considered. We show how the higher dimensional operators can be written as sums of tensor products of one dimensional operators, and th...

A new positive definite expanded mixed finite element method is proposed for parabolic partial integrodifferential equations. Compared to expanded mixed scheme, the new expanded mixed element system is symmetric positive definite and both the gradient equation and the flux equation are separated from its scalar unknown equation. The existence and uniqueness for semidiscrete scheme are proved an...

The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in L2. Any standard mixed finite element function space can be utilized for this mixed formulation, e.g., the Raviart-Thomas discretization which is r...

We are concerned with the finite-element approximation for the Keller-Segel system that describes the aggregation of slime molds resulting from their chemotactic features. The scheme makes use of a semi-implicit time discretization with a time-increment control and Baba-Tabata’s conservative upwind finite-element approximation in order to realize the positivity and mass conservation properties....

In this article, we consider a fully discrete stabilized finite element method based on two local Gauss integrations for the two-dimensional time-dependent Navier–Stokes equations. It focuses on the lowest equal-order velocity–pressure pairs. Unlike the other stabilized method, the present approach does not require specification of a stabilization parameter or calculation of higher-order deriva...

explosion has always been regarded as one of the most complicated engineering problems. as a result, many engineers have preferred rather simplified empirical approaches in comparison to extremely complex deterministic analyses. in this paper, however, a numerical simulation based on the combined finite/discrete element methodology is presented for analyzing the dynamic behavior of fracturing r...

In [Z. Cai, C. Tong, P. S. Vassilevski, and C. Wang, Numer. Methods Partial Differential Equations, to appear], the authors developed and analyzed a mixed finite element method for the stationary Stokes equations based on the pseudostress-velocity formulation. The pseudostress and the velocity are approximated by a stable pair of finite elements: Raviart–Thomas elements of index k ≥ 0 and disco...

This paper is about discrete sensitivity analysis. A triangular bending element with constant moment and six degrees of freedom is used. The required derivatives for sensitivity analysis are calculated explicitly. These formulations, finite element method and sequential linear programming are utilized to find shape optimization of plate bending structures. The numerical examples, which show the...

Stability and convergence analyses for the domain decomposition finite element/finite difference (FE/FD) method are presented. The developed a semi-discrete element scheme time-dependent Maxwell’s equations. explicit schemes in different settings of spatial constructed algorithm is formulated. Several numerical examples validate rates obtained theoretical studies.