We investigate novel fitted finite element approximations for two-phase Navier–Stokes flow. In particular, we consider both Eulerian and Arbitrary Lagrangian–Eulerian (ALE) finite element formulations. The moving interface is approximated with the help of parametric piecewise linear finite element functions. The bulk mesh is fitted to the interface approximation, so that standard bulk finite element spaces can be used throughout. The meshes describing the discrete interface in general do not deteriorate in time, which means that in numerical simulations a smoothing or a remeshing of the interface mesh is not necessary. We present several numerical experiments, including convergence experiments and benchmark computations, for the introduced numerical methods, which demonstrate the accuracy and robustness of the proposed algorithms. We also compare the accuracy and efficiency of the Eulerian and ALE formulations.

In this article, we consider a weak Galerkin finite element method for the two dimensional exterior Helmholtz problem. After introducing a nonlocal boundary condition by means of the exact Dirichlet to Neumann (DtN) operator for the exterior problem, we prove that the existence and uniqueness of the weak Galerkin finite element solution for this problem. Then, applying some projection techniques, we establish a priori error estimate, which include the effect of truncation of the DtN boundary condition as well as the spatial discretization. Finally, some numerical examples are presented to confirm the theoretical predictions.

In this paper, we design a class of general split-step methods for solving Itô stochastic differential systems, in which the drift or deterministic increment function can be taken from special ordinary differential equations solver, based on the harmonic-mean. This method is justified to have a strong convergence order of $\frac{1}{2}$. Further, we investigate mean-square stability of the proposed method for linear scalar stochastic differential equation. Finally, some examples are included to demonstrate the validity and efficiency of the introduced scheme.

In this paper, we post-process the finite element solutions for parabolic equations to meet discrete conservation laws in element-level. The post-processing procedure are implemented by two different approaches: one is by computing a globally continuous flux function and the other is by computing the so-called finite-volume-element-like solution. Both approaches only require to solve a small linear system on each element of the underlying mesh. The post-processed flux converges to the exact flux with optimal convergence rates. Numerical computations verify our theoretical findings.

A new interior-exterior penalty method for solving quasi-variational inequality and pseudo-monotone operator arising in two-dimensional point contact problem is analyzed and developed in discontinuous Galerkin finite volume (DG-FVEM) framework. We derive a discrete DG-FVEM formulation of the problem and prove existence and uniqueness results for it. Optimal error estimates in $H^1$ and $L^2$ norm are derived under a light load parameter assumptions. In addition, the article provides a complete algorithm to tackle all numerical complexities appear in the solution procedure. Numerical outcomes are presented for light, moderate and relative high load conditions. The variations of load parameter and its effect on the evolution of deformations and pressure profile are evaluated and described. This method is well suited for solving elasto-hydrodynamic lubrication point contact problems and can probably be treated as commercial software. Furthermore, the results give a hope for the further development of the scheme for extreme load condition, observations in a more realistic operating situation which will be discussed in part II.

In this paper, we solve the time-dependent Stokes problem by the weak Galerkin (WG) finite element method. Full-discrete WG finite element scheme is obtained by applying the implicit backward Euler method for time discretization. Optimal order error estimates are established for the corresponding numerical approximation in $H^1$ norm for the velocity, and $L^2$ norm for both the velocity and the pressure in semi-discrete forms and full-discrete forms, respectively. Some computational results are presented to demonstrate the accuracy, convergence and efficiency of the method.

Aeroelastic analysis of the aircraft is a typical fluid-structure interaction problem. It is influenced by interactions between aerodynamic forces and deformations of elastic structures. The aerodynamic field and structural deformation are modeled by different physical equations, and the associated computational meshes do not match each other. Therefore, passing data from a mesh to the other one in a physically reasonable way is a challenging task. Current aerodynamic force transportation methods, such as virtual work conserved method (VWC), area weighted shape function method (AWSF), proximity minimum strain energy method (PMSE), and inverse distance weighted method (IDW), either destroy physical conservations or cause unreasonable distributions of structural forces. In this paper we propose a corrected nearest neighbor transportation method (CNNT) of aerodynamic force for the fluid-structure coupling analysis. The force transportation process is divided into two phases. First, the aerodynamic forces are allocated to the structural nodes initially using the conventional methods or, e.g., AWSF, IDW. Then, the initially allocated structural forces are corrected by solving an optimization problem with the physical conservations as its optimization target. The optimization problem is solved by a barrier interior point method efficiently. A sport airplane model is employed to verify effectiveness of CNNT. Comparisons with the VWC, AWSF, PMSE, IDW are also made. The numerical experiments show that the CNNT maintains the force, moment, and virtual work conservations, and exhibits reasonable distributions of structural forces, indeed.