How to properly specify boundary conditions for pressure is a longstanding problem for the incompressible Navier–Stokes equations with no-slip boundary conditions. An analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the Laplacian and Leray projection operators. Here we make use of this formula to (a) improve the accuracy of computing pressure in two kinds of existing time-discrete projection methods implicit in viscosity only, and (b) devise new higher-order accurate time-discrete projection methods that extend a slip-correction idea behind the well-known finite-difference scheme of Kim and Moin. We test these schemes for stability and accuracy using various combinations of C0 finite elements. For all three kinds of time discretization, one can obtain third-order accuracy for both pressure and velocity without a time-step stability restriction of diffusive type. Furthermore, two kinds of projection methods are found stable using piecewise-linear elements for both velocity and pressure.

In this paper, we propose and analyze a new stabilized finite element method using continuous piecewise linear (or bilinear) elements for solving 2D reaction-convection-diffusion equations. The equation under consideration involves a small diffusivity $\varepsilon$ and a large reaction coefficient $\sigma$, leading to high P\'eclet number and high Damk\"{o}hler number. In addition to giving error estimates of the approximations in $L^2$ and $H^1$ norms, we explicitly establish the dependence of error bounds on the diffusivity, the $L^{\infty}$ norm of convection field, the reaction coefficient and the mesh size. Our analysis shows that the proposed method is particularly suitable for problems with a small diffusivity and a large reaction coefficient, or more precisely, with a large mesh P\'eclet number and a large mesh Damk\"{o}hler number. Several numerical examples exhibiting boundary or interior layers are given to illustrate the high accuracy and stability of the proposed method. The results obtained are also compared with those of existing stabilization methods.

H(div) conforming and discontinuous Galerkin (DG) methods are designed for incompressible Euler's equation in two and three dimension. Error estimates are proved for both the semi-discrete method and fully-discrete method using backward Euler time stepping. Numerical examples exhibiting the performance of the methods are given.

We propose a computational efficient yet simple numerical algorithm to solve the surface eikonal equation on general implicit surfaces. The method is developed based on the embedding idea and the fast sweeping methods. We first approximate the solution to the surface eikonal equation by the Euclidean weighted distance function defined in a tubular neighbourhood of the implicit surface, and then apply the efficient fast sweeping method to numerically compute the corresponding viscosity solution. Unlike some other embedding methods which require the radius of the computational tube satisfies h = O ( x ) for some h = O ( x ) , our approach allows h = O ( x ) . This implies that the total number of grid points in the computational tube is optimal and is given by h = O ( x ) for a co-dimensional one surface in h = O ( x ) . The method can be easily extended to general static HamiltonJacobi equation defined on implicit

In this paper, we develop an energy stable fully discrete discontinuous Galekin (DG) finite element method for the thin film epitaxy problem. Based on the method of lines, we construct and prove the energy stability of the spatial semi-discrete DG scheme firstly. To avoid the strict time step restriction of the explicit time integration method, the first order convex splitting method is used to get an unconditionally stable fully discrete DG method, which is a linearly implicit scheme for this nonlinear problem. The energy stability of the fully discrete convex splitting DG scheme is also proved. To improve the temporal accuracy, spectral deferred correction (SDC) method is adapted to achieve the high order accuracy in both time and space. Combining with the convex splitting method, the SDC method can be linearly solvable, high order accurate and stable in our numerical tests. These advantages ensure that the resulting fully discrete DG scheme is efficient to perform the long time simulation of the thin film epitaxy model. Numerical experiments of the accuracy and long time simulation show the capability and efficiency of the method.