In this article we define a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\Gamma$ in $\mathbb R^{n+1}$. The key idea is based on the approximation of $\Gamma$ by a polyhedral surface $\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices is simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. Numerical experiments are described for several linear and nonlinear partial differential equations. In particular, the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow.

In this paper, three $n$-rectangle nonconforming elements are proposed with $n\ge3$. They are the extensions of well-known Morley element, Adini element and Bogner-Fox-Schmit element in two spatial dimensions to any higher dimensions respectively. These elements are all proved to be convergent for a model biharmonic equation in $n$ dimensions.

The fast sweeping method is an efficient iterative method for hyperbolic problems. It combines Gauss-Seidel iterations with alternating sweeping orderings. In this paper several parallel implementations of the fast sweeping method are presented. These parallel algorithms are simple and efficient due to the causality of the underlying partial different equations. Numerical examples are used to verify our algorithms.

In this paper, two new nonconforming hexagonal elements are presented, which are based on the trilinear function space $Q_1^{(3)}$ and are edge-oriented, analogical to the case of the rotated $Q_1$ quadrilateral element. A priori error estimates are given to show that the new elements achieve first-order accuracy in the energy norm and second-order accuracy in the $L^2$ norm. This theoretical result is confirmed by the numerical tests.

We construct and implement a non-oscillatory relaxation scheme for multidimensional hyperbolic systems of conservation laws. The method transforms the nonlinear hyperbolic system to a semilinear model with a relaxation source term and linear characteristics which can be solved numerically without using either Riemann solver or linear iterations. To discretize the relaxation system we consider a high-resolution reconstruction in space and a TVD Runge-Kutta time integration. Detailed formulation of the scheme is given for problems in three space dimensions and numerical experiments are implemented in both scalar and system cases to show the effectiveness of the method.

Consider the diffraction of a time-harmonic wave incident upon a periodic chiral structure. The diffraction problem may be simplified to a two-dimensional one. In this paper, the diffraction problem is solved by a finite element method with perfectly matched absorbing layers (PMLs). We use the PML technique to truncate the unbounded domain to a bounded one which attenuates the outgoing waves in the PML region. Our computational experiments indicate that the proposed method is efficient, which is capable of dealing with complicated chiral grating structures.

We estimate error bounds between ternary subdivision curves/surfaces and their control polygons after $k$-fold subdivision in terms of the maximal differences of the initial control point sequences and constants that depend on the subdivision mask. The bound is independent of the process of subdivision and can be evaluated without recursive subdivision. Our technique is independent of parametrization therefore it can be easily and efficiently implemented. This is useful and important for pre-computing the error bounds of subdivision curves/surfaces in advance in many engineering applications such as surface/surface intersection, mesh generation, NC machining, surface rendering and so on.

In this paper, we consider a hydrodynamic model of the semiconductor device. The approximate solutions are obtained by a mixed finite volume method for the potential equation and multistep upwind finite volume methods for the concentration equations. Error estimates in some discrete norms are derived under some regularity assumptions on the exact solutions.