Volume 29, Issue 2

A finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly thewetting phase pressure and saturation, which are the primary unknowns. The discrete saturation satisfies a maximum principle. Stability of the scheme and existence of a solution are established.

We propose a Discontinuous Galerkin (DG) scheme for the hydrostatic Stokes equations. These equations, related to large-scale PDE models in oceanography, are characterized by the loss of ellipticity of the vertical momentum equation. This fact provides some interesting challenges, such as the design of stable numerical schemes. The new scheme proposed here is based on the symmetric interior penalty (SIP) technique, with a particular treatment of the vertical velocity. It is well-known that stability of the mixed formulation of primitive equations requires, besides the LBB inf-sup condition, an additional hydrostatic inf-sup restriction relating pressure and vertical velocity. This hydrostatic inf-sup condition invalidates stability of usual Stokes stable continuous finite elements like Taylor-Hood 𝓟 2 /𝓟 1 or bubble 𝓟 1 b /𝓟 1 . Here we prove stability for our 𝓟 k /𝓟 k DG scheme. Some novel numerical tests are provided which are in agreement with the previous analysis.

In this paper, we study the relative error in the numerical solution of a linear ordinary differential equation y '( t ) = Ay ( t ), t ≥ 0, where A is a normal matrix. The numerical solution is obtained by using at any step an approximation of the matrix exponential, e.g., a polynomial or a rational approximation. The error of the numerical solution with respect to the exact solution is due to this approximation as well as to a possible perturbation in the initial value. For an unperturbed initial value, we have found: (1) unlike the absolute error, the relative error always grows linearly in time; (2) in the long-time, the contributions to the relative error relevant to non-rightmost eigenvalues of A disappear.

For nonlinear nonsmooth DC programming (difference of convex functions), we introduce a new redistributed proximal bundle method. The subgradient information of both the DC components is gathered from some neighbourhood of the current stability center and it is used to build separately an approximation for each component in the DC representation. Especially we employ the nonlinear redistributed technique to model the second component of DC function by constructing a local convexification cutting plane. The corresponding convexification parameter is adjusted dynamically and is taken sufficiently large to make the `augmented' linearization errors nonnegative. Based on above techniques we obtain a new convex cutting plane model of the original objective function. Based on this new approximation the redistributed proximal bundle method is designed and the convergence of the proposed algorithm to a Clarke stationary point is proved. A simple numerical experiment is given to show the validity of the presented algorithm.