In this paper, we present a discontinuous Galerkin (DG) method based on the Nédélec finite element space for solving a fourth-order curl equation arising from a magnetohydrodynamics model on a 3-dimensional bounded Lipschitz polyhedron. We show that the method has an optimal error estimate for a model problem involving a fourth-order curl operator. Furthermore, some numerical results in 2 dimensions are presented to verify the theoretical results.

In this paper, we investigate spectral method for mixed inhomogeneous boundary value problems in three dimensions. Some results on the three-dimensional Legendre approximation in Jacobi weighted Sobolev space are established, which improve and generalize the existing results, and  play an important role in numerical solutions of partial differential equations. We also develop a lifting technique, with which we could handle mixed inhomogeneous boundary conditions easily. As examples of applications, spectral schemes are provided for three model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms is proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy, and confirm the theoretical analysis well.

We consider the inverse scattering problem for scalar waves. We analyze the convergence of the inverse Born series and study its use in numerical simulations for the case of a spherically-symmetric medium in two and three dimensions.

We construct a new stabilized finite volume method on rectangular grids for the Stokes equations. The lowest equal-order conforming finite element pair (piecewise bilinear velocities and pressures) and piecewise constant test spaces for both the velocity and pressure are employed in this method. We show the stability of this method and prove first optimal rate of convergence for the velocity in the $H^1$ norm and the pressure in the $L^2$ norm. In addition, a second order optimal error estimate for the velocity in the $L^2$ norm is derived. Numerical experiments illustrating the theoretical results are included.

In this paper, we propose adaptive finite element methods with error control for solving elasticity problems with discontinuous coefficients. The meshes in the methods do not need to fit the interfaces. We establish a residual-based a posteriori error estimate which is $λ$-independent multiplicative constants; the Lamé constant $λ$ steers the incompressibility. The error estimators are then implemented and tested with promising numerical results which will show the competitive behavior of the adaptive algorithm.

This paper is concerned with the initial-boundary value problems of scalar transport equations with uncertain transport velocities. It was demonstrated in our earlier works that regularity of the exact solutions in the random spaces (or the parametric spaces) can be determined by the given data. In turn, these regularity results are crucial to convergence analysis for high order numerical methods. In this work, we will prove the spectral convergence of the stochastic Galerkin and collocation methods under some regularity results or assumptions. As our primary goal is to investigate the errors introduced by discretizations in the random space, the errors for solving the corresponding deterministic problems will be neglected.

The error caused by the ghost force is studied for a quasicontinuum method with planar interface in two dimension. For a special case, we derive an analytical expression of the error, which is exploited to prove that the ghost force may lead to a finite size error for the gradient of the solution. The pointwise estimate of the error shows that the error decays algebraically away from the interface, which is much slower than that of the one-dimensional problem, for which the error decays exponentially away from the interface.