Discrete maximum principles are proved for finite element discretizations of nonlinear elliptic interface problems with jumps of the normal derivatives. The geometric conditions in the case of simplicial meshes are suitable acuteness or nonobtuseness properties.

In this paper, the Lamé coefficients in the linear elasticity problem are estimated by using the measurements of displacement. Some a posteriori error estimators for the approximation error of the parameters are derived, and then adaptive finite element schemes are developed for the discretization of the parameter estimation problem, based on the error estimators. The Gauss-Newton method is employed to solve the discretized nonlinear least-squares problem. Some numerical results are presented.

We propose a stable, convergent finite difference scheme to solve numerically a nonlocal Allen-Cahn equation which may model a variety of physical and biological phenomena involving long-range spatial interaction. We also prove that the scheme is uniquely solvable and the numerical solution will approach the true solution in the $L^∞$ norm.

We are concerned in this work with simulations of the localization of a finite number of small electromagnetic inhomogeneities contained in a three-dimensional bounded domain. Typically, the underlying inverse problem considers the time-harmonic Maxwell equations formulated in electric field in this domain and attempts, from a finite number of boundary measurements, to localize these inhomogeneities. Our simulations are based on an approach that combines an asymptotic formula for perturbations in the electromagnetic fields, a suited inversion process, and finite element meshes derived from a non-standard discretization process of the domain. As opposed to a recent work, where the usual discretization process of the domain was employed in the computations, here we localize inhomogeneities that are one order of magnitude smaller.

We use the modified method of characteristics for solving nonlinear convection diffusion problems with memory terms. The convergence of approximation scheme is proved under minimal regularity assumptions on the velocity field and on the solution. The results are supported by numerical experiments for contaminant transport with diffusion and non-equilibrium sorption isotherms.

In this paper we develop two novel numerical methods for the partial integral differential equation arising from the valuation of an option whose underlying asset is governed by a jump diffusion process. These methods are based on a fitted finite volume method for the spatial discretization, an implicit-explicit time stepping scheme and the Crank-Nicolson time stepping method. We show that the discretization methods are unconditionally stable in time and the system matrices of the resulting linear systems are M-matrices. The resulting linear systems involve products of a dense matrix and vectors and an Fast Fourier Transformation (FFT) technique is used for the evaluation of these products. Furthermore, a splitting technique is proposed for the solution of the discretized system arising from the Crank-Nicolson scheme. Numerical results are presented to show the rates of convergence and the robustness of the numerical method.

In this paper we establish the stability and convergence of the time-domain perfectly matched layer (PML) method for solving the acoustic scattering problems. We first prove the well-posedness and the stability of the time-dependent acoustic scattering problem with the Dirichlet-to-Neumann boundary condition. Next we show the well-posedness of the unsplit-field PML method for the acoustic scattering problems. Then we prove the exponential convergence of the non-splitting PML method in terms of the thickness and medium property of the artificial PML layer. The proof depends on a stability result of the PML system for constant medium property and an exponential decay estimate of the modified Bessel functions.

In this paper, we are concerned with a non-overlapping domain decomposition method with nonmatching grids. In this method, a new pointwise matching condition is used to define weak continuity of approximate solutions on the interface. The main merit of the new method is that numerical integrations can be avoided when calculating interface matrices. We derive an almost optimal error estimate of the resulting approximate solutions for two kinds of applicable situations. Some numerical experiments confirm the theoretical result.

We study the dual mixed finite element approximation of unilateral contact problems. Based on the dual mixed variational formulation with three unknowns (stress, displacement and the displacement on the contact boundary), the a priori error estimates have been established for both conforming and nonconforming finite element approximations. A Uzawa type iterative algorithm is developed to solve the resulting linear system. Numerical example shows good performance of the method.