In this work, a stationary 2d Navier-Stokes problem with nonlinear feedback forces field is considered in the stream-function formulation. We use the Continuous/Discontinuous Finite Element Method (CD-FEM), with interior penalty terms, to numerically solve the associated boundary-value problem. For the associated continuous and discrete problems, we prove the existence of weak solutions and establish the conditions for their uniqueness. Consistency, stability and convergence of the method are also shown analytically. To validate the numerical model regarding its applicability and robustness, several test cases are carried out.

This paper presents a direct method for efficiently solving three-dimensional elliptic interface problems featuring piecewise constant coefficients with a finite jump across the interface. A key advantage of our approach lies in its avoidance of augmented variables, distinguishing it from traditional methods. The computational framework relies on a finite difference scheme implemented on a uniform Cartesian grid system. By utilizing a seven-point Laplacian for grid points away from the interface, our method only requires coefficient modifications for grid points located near or on the interface. Numerical experiments validate our method’s effectiveness. Generally, it achieves second-order accuracy for both the solution and its gradient, measured in the maximum norm, particularly effective in scenarios with moderate coefficient jumps. Extending and building upon the recent work of [1] on 1D and 2D elliptic interfaces, our approach successfully introduces a simpler method for extension into three dimensions. Notably, our proposed method not only offers efficiency and accuracy but also enhances the simplicity of implementation, making it accessible to non-experts in the field.

This paper presents a recently developed full kinetic particle simulation code package, which is a two-dimensional highly integrated and universal framework for low-temperature plasma simulation on both Cartesian and axisymmetric coordinate systems. This code package is named CFIRM, since it is designed based on the continuous Galerkin immersed-finite-element (IFE) particle-in-cell (PIC) model with the polynomial-preserving-recovery (PPR) technique and the Monte-Carlo-collision (MCC) method. Both the traditional and implicit PIC methods were implemented in the package. Incorporating the advantages of all these methods together, the CFIRM code can adopt explicit or implicit PIC schemes to track the motion trajectory of charged particles and deal with the collisions between plasma and neutral gas. Additionally, it can conveniently handle complex interface problems on structured grids. The CFRIM code has excellent versatility in low-temperature plasma simulation and can easily extend to various particle processing modules, such as the variable weights and adaptive particle management algorithms which were incorporated into this code to reduce the memory utilization rate. The implementation for the main algorithms and the overall simulation framework of the CFIRM code package are rigorously described in details. Several simulations of the benchmark cases are carried out to validate the reliability and accuracy of the CFIRM code. Moreover, two typical low-temperature plasma engineering problems are simulated, including a hall thruster and a capacitively coupled plasma reactor, which demonstrates the applicability of this code package.

In this paper, we consider 3D steady convection-diffusion equations in cylindrical domains. Instead of applying the finite difference methods (FDM) or the finite element methods (FEM), we propose a difference finite element method (DFEM) that can maximize good applicability and efficiency of both FDM and FEM. The essence of this method lies in employing the centered difference discretization in the $z$-direction and the finite element discretization based on the $P_1$ conforming elements in the $(x, y)$ plane. This allows us to solve partial differential equations on complex cylindrical domains at lower computational costs compared to applying the 3D finite element method. We derive stability estimates for the difference finite element solution and establish the explicit dependence of $H_1$ error bounds on the diffusivity, convection field modulus, and mesh size. Finally, we provide numerical examples to verify the theoretical predictions and showcase the accuracy of the considered method.

This paper develops novel fast numerical solvers for subdiffusion problems with spatial interfaces. These problems are modeled by partial differential equations that contain both fractional order and conventional first order time derivatives. The former is non-local and approximated by L1 and L2 discretizations along with fast evaluation algorithms based on approximation by sums of exponentials. This results in an effective treatment of the “long-tail” kernel of subdiffusion. The latter is local and hence conventional implicit Euler or Crank-Nicolson discretizations can be used. Finite volumes are utilized for spatial discretization based on consideration of local mass conservation. Interface conditions for mass and fractional fluxes are incorporated into these fast solvers. Computational complexity and implementation procedures are briefly discussed. Numerical experiments demonstrate accuracy and efficiency of these new fast solvers.