An $H^1$-Galerkin mixed finite element method is discussed for a class of second order hyperbolic problems. It is proved that the Galerkin approximations have the same rates of convergence as in the classical mixed method, but without LBB stability condition and quasi-uniformity requirement on the finite element mesh. Compared to the results proved for one space variable, the $L^∞(L^2)$-estimate of the stress is not optimal with respect to the approximation property for the problems in two and three space dimensions. It is further noted that if the Raviart- Thomas spaces are used for approximating the stress, then optimal estimate in $L^∞(L^2)$-norm is achieved using the new formulation. Finally, without restricting the approximating spaces for the stress, a modification of the method is proposed and analyzed. This confirms the findings in a single space variable and also improves upon the order of convergence of the classical mixed procedure under an extra regularity assumption on the exact solution.

In this paper an explicit/implicit schemes for parabolic equations with discontinuous coefficients is analyzed. We show that the error of the solution in $L^∞$ norm and the error of the discrete flux in $L^2$ norm are in order $O(\tau + h^2)$ and $O(\tau + h^{\frac{3}{2}})$, respectively and the scheme is stable under some weaker conditions, while the difference scheme has the truncation error $O(1)$ at the neighboring points of the discontinuity of the coefficient. Numerical experiments, which are given for both linear and nonlinear problems, show that our theoretical estimates are optimal in some sense. The comparison with some classical scheme is presented.

Practical QR algorithm for the real unsymmetric algebraic eigenvalue problem is considered. The global convergence of shifted QR algorithm in finite precision arithmetic is addressed based on a model of the dynamics of QR algorithm in a neighborhood of an unreduced Hessenberg fixed point. The QR algorithm fails at a "stable" unreduced fixed point. Prior analyses have either determined some unstable unreduced Hessenberg fixed points or have addressed stability to perturbations of the reduced Hessenberg fixed points. The model states that sufficient criteria for stability (e.g. failure) in finite precision arithmetic are that a fixed point be neutrally stable both with respect to perturbations that are constrained to the orthogonal similarity class and to general perturbations from the full matrix space. The theoretical analysis presented herein shows that at an arbitrary unreduced fixed point "most" of the eigenvalues of the Jacobian(s) are of unit modulus. A framework for the analysis of special cases is developed that also sheds some light on the robustness of the QR algorithm.

A level set method is proposed for solving free boundary problems coming from contact with obstacles. Two different approaches are described and applied for solving a unilateral obstacle problem. The cost functionals coming from the new approach are nonsmooth. For solving the nonsmooth minimization problems, two methods are applied: firstly, a proximal bundle method, which is a method for solving general nonsmooth optimization problems. Secondly, a gradient method is proposed for solving the regularized problems. Numerical experiments are included to verify the convergence of the methods and the quality of the results.

A system of jump-diffusion stochastic differential equations is considered for modelling the dynamics of the spread of an amphibian disease. In this investigation, it is assumed that the amphibians are located in $M$ regions which are widely and uniformly spaced on the surface of the earth and that the disease is present initially in only one region. Within each region, the amphibians live in $N$ separate patches. A jump-diffusion stochastic system is derived for the number of infected patches in each of the $M$ regions. Computational simulations are performed and compared with results predicted by a deterministic SIS model, a continuous-trajectory stochastic differential equation model, and Monte Carlo calculations. It is seen that the rate of spread predicted by the jump-diffusion model agrees well with that predicted by Monte Carlo calculations. Indeed, if there is a step increase in the transmission rates or a step decrease in the recovery rates, then the disease can spread globally from region to region at an exponential rate.

The simulation of realistic multiphase flow problems in petroleum reservoirs requires means for handling the complicated structure of the reservoirs such as complex boundaries, faults, fractures, and horizontal wells. A numerical reservoir simulator has recently been developed to be able to handle these features for a wide range of applications. This fully implicit simulator is based on a three-dimensional, three-phase black oil model. It can also be used to solve a dual-porosity, dual-permeability black oil model in a fractured reservoir. The space discretization method used in this simulator is based on a block-centered finite difference method with harmonic averaged coefficients (equivalently, a mixed finite element method). In this paper we report an application of this simulator to a problem involving injection and production from horizontal wells in a reservoir where a coning tendency is important, and present a benchmark comparison with other simulators by fourteen petroleum organizations.

In this paper we study American put options on zero-coupon bonds under the CIR model of short interest rates. The uniqueness of the optimal exercise boundary and the solution existence and uniqueness of a degenerate parabolic free boundary problem are established. Numerical examples are also presented to confirm theoretical results.