In this paper, the magneto-heating model is considered, where the nonlinear terms conclude the coupling magnetic diffusivity, the turbulent convection zone, the flow fields, ohmic heat, and α-quench. The highlights of this paper consist of three parts. Firstly, the solvability of the model is derived from Rothe's method and Arzela-Ascoli theorem after setting up the decoupled semi-discrete system. Secondly, the well-posedness for the full-discrete scheme is arrived and the convergence order $O$($h$^min{$s$,$m$}+$τ$) is obtained, respectively, where the approximation scheme is based on backward Euler discretization in time and Nédélec-Lagrangian finite elements in space. At last, a numerical experiment demonstrates the expected convergence.

This paper investigates the long time stability behavior of multiphysics flow problems, namely the Navier-Stokes equations, natural convection and double-diffusive convection equations with an extrapolated blended BDF time-stepping scheme. This scheme combines the two-step BDF and three-step BDF time stepping schemes. We prove unconditional long time stability theorems for each of these flow systems. Various numerical tests are given for large time step sizes in long time intervals in order to support theoretical results.

In this paper, a branch of nonsingular solutions of the stationary Navier-Stokes equations are investigated, which are unique on a neighborhood, and mostly isolated without relying on very stringent requirement on the small data. We summarize and develop an equivalent definition of nonsingular solutions of finite element/finite volume methods in the same framework. Furthermore, we establish the equivalent definition of a branch of singular solutions of finite element methods for the coupled Navier-Stokes/Darcy equations.

We construct a family of conforming piecewise harmonic finite elements on triangulations. Because the dimension of harmonic polynomial spaces of degree ≤ $k$ is much smaller than the one of the full polynomial space, the triangles in the partition must be refined in order to achieve optimal order of approximation power. We use the Hsieh-Clough-Tocher split: the barycenter of each original triangle is connected to its three vertices. Depending on the polynomial degree $k$, the original triangles have some minor restrictions which can be easily fulfilled by small perturbations of some vertices of the original triangulation. The optimal order of convergence is proved for the conforming harmonic finite elements, and confirmed by numerical computations. Numerical comparisons with the standard finite elements are presented, showing advantages and disadvantages of the harmonic finite element method.

This paper is concerned with the global stability of non-critical/critical traveling waves with oscillations for time-delayed nonlocal dispersion equations. We first theoretically prove that all traveling waves, especially the critical oscillatory traveling waves, are globally stable in a certain weighted space, where the convergence rates to the non-critical oscillatory traveling waves are time-exponential, and the convergence to the critical oscillatory traveling waves are time-algebraic. Both of the rates are optimal. The approach adopted is the weighted energy method with the fundamental solution theory for time-delayed equations. Secondly, we carry out numerical computations in different cases, which also confirm our theoretical results. Because of oscillations of the solutions and nonlocality of the equation, the numerical results obtained by the regular finite difference scheme are not stable, even worse to be blow-up. In order to overcome these obstacles, we propose a new finite difference scheme by adding artificial viscosities to both sides of the equation, and obtain the desired numerical results.

In this paper, we propose and analyze a decoupled second order backward difference formula (BDF2) time-stepping algorithm for solving transient viscoelastic fluid flow. The spatial discretization is based on continuous Galerkin finite element approximation for the velocity and pressure, and discontinuous Galerkin finite element approximation for the viscoelastic stress tensor. To obtain a non-iterative decoupled algorithm from the fully discrete nonlinear system, we employ a second order extrapolation in time to the nonlinear terms. The algorithm requires the solution of one Navier-Stokes problem and one constitutive equation per time step. For mesh size $h$ and temporal step size ∆$t$ sufficiently small satisfying ∆$t$ ≤ $Ch$^$d$/4, a priori error estimates in terms of ∆$t$ and $h$ are derived. Numerical tests are presented that illustrate the accuracy and stability of the algorithm.

A new finite element method with discontinuous approximation is introduced for solving second order elliptic problem. Since this method combines the features of both conforming finite element method and discontinuous Galerkin (DG) method, we call it conforming DG method. While using DG finite element space, this conforming DG method maintains the features of the conforming finite element method such as simple formulation and strong enforcement of boundary condition. Therefore, this finite element method has the flexibility of using discontinuous approximation and simplicity in formulation of the conforming finite element method. Error estimates of optimal order are established for the corresponding discontinuous finite element approximation in both a discrete $H$¹ norm and the $L$² norm. Numerical results are presented to confirm the theory.

We proposed a 3D $B$₂ model for the radiative transfer equation. The model is an extension of the 1D $B$₂ model for the slab geometry. The 1D $B$₂ model is an approximation to the 2nd order maximum entropy ($M$₂) closure and has been proved to be globally hyperbolic. In 3D space, we are basically following the method for the slab geometry case to approximate the $M$₂ closure by $B$₂ ansatz. Same as the $M$₂ closure, the ansatz of the new 3D $B$₂ model has the capacity to capture both isotropic solutions and strongly peaked solutions. And beyond the $M$₂ closure, the new model has fluxes in closed-form such that it is applicable to practical numerical simulations. The rotational invariance, realizability, and hyperbolicity of the new model are carefully studied.