We present discrete energy decay results for the Yee scheme applied to Maxwell's equations in Debye and Lorentz dispersive media. These estimates provide stability conditions for the Yee scheme in the corresponding media. In particular, we show that the stability conditions are the same as those for the Yee scheme in a nondispersive dielectric. However, energy decay for the Maxwell-Debye and Maxwell-Lorentz models indicate that the Yee schemes are dissipative. The energy decay results are then used to prove the convergence of the Yee schemes for the dispersive models. We also show that the Yee schemes preserve the Gauss divergence laws on its discrete mesh. Numerical simulations are provided to illustrate the theoretical results.

In this work, we study the finite element approximation of a model optimal control problem governed by the system describing the two-phase incompressible flow in porous media, with the aim to maximize production of oil from petroleum reservoirs. We first give the proof for the existence of the solutions of the control problem. The optimality conditions are then obtained and the existence of the solution of the adjoint equations is shown. After that we consider its finite element approximation. We have obtained the a priori error estimates with the optimal orders and minimum regularity requirements. Finally, we carry out some numerical tests.

The discontinuous Galerkin Interior Penalty Method with solenoidal approximations proposed in [13] for the incompressible Stokes equations is analyzed. Continuity and coercivity of the bilinear form are proved. A priori error estimates, with optimal convergence rates, are derived. 2D and 3D numerical examples with known analytical solution corroborate the theoretical analysis.

In this work, numerical schemes to approximate the solution of one and multi phase quadrature domains are presented. We shall construct a monotone, stable and consistent finite difference method for both one and two phase cases, which converges to the viscosity solution of the partial differential equation arising from the corresponding quadrature domain theory. Moreover, we will discuss the numerical implementation of the resulting approach and present computational tests.

We analyze the convergence of a mixed finite element method for the elliptic Monge-Ampère  equation in dimensions 2 and 3. The unknowns in the formulation, the scalar variable and a discrete Hessian, are approximated by Lagrange finite element spaces. The method originally proposed by Lakkis and Pryer can be viewed as the formal limit of a Hermann-Miyoshi mixed method proposed by Feng and Neilan in the context of the vanishing moment methodology. Error estimates are derived under the assumption that the continuous problem has a smooth solution.

We consider a simple advection equation in space dimension one and the linearized shallow water equations in space dimension two and describe and implement two different multilevel finite volume discretizations in the context of the utilization of the incremental methods with time explicit or semi-explicit schemes.

In this paper, we develop pollution-free finite difference schemes for solving the non-homogeneous Helmholtz equation in one dimension. A family of high-order algorithms is derived by applying the Taylor expansion and imposing the conditions that the resulting finite difference schemes satisfied the original equation and the boundary conditions to certain degrees. The most attractive features of the proposed schemes are: first, the new difference schemes have a $2n$-order of rate of convergence and are pollution-free. Hence, the error is bounded even for the equation at high wave numbers. Secondly, the resulting difference scheme is simple, namely it has the same structure as the standard three-point central differencing regardless of the order of accuracy. Convergence analysis is presented, and numerical simulations are reported for the non-homogeneous Helmholtz equation with both constant and varying wave numbers. The computational results clearly confirm the superior performance of the proposed schemes.

In this article, we consider the linearized inviscid shallow water equations in space dimension two in a rectangular domain. We implement a finite volume discretization and prove the numerical stability and convergence of the scheme for three cases that depend on the background flow $\tilde{u}_0$, $\tilde{v}_0$, and $\tilde{\phi}_0$ (sub- or super-critical flow at each part of the boundary). The three cases that we consider are fully hyperbolic modes.

The scattering problem from time-harmonic waves by a Neumann type crack in $\mathbb{R}^2$ is considered. A PML technique is used for solving the problem with a bounded domain instead of the infinite domain. A coupled finite-infinite element method is employed in the computation. Because of the singularity of the solution, the infinite element method is used near the crack tip. An error analysis is presented for the numerical approximation. The convergence order of the method is higher than FEM's.

A parallel variational multiscale method based on the partition of unity is proposed for incompressible flows in this paper. Based on two-grid method, this algorithm localizes the global residual problem of variational multiscale method into a series of local linearized residual problems. To decrease the undesirable effect of the artificial homogeneous Dirichlet boundary condition of local sub-problems, an oversampling technique is also introduced. The globally continuous finite element solutions are constructed by assembling all local solutions together using the partition of unity functions. Numerical simulations demonstrate the high efficiency and flexibility of the new algorithm.