We consider the coupling of compressible and nearly incompressible materials within the framework of mortar methods. Taking into account the locking effect, we use a suitable discretization for the nearly incompressible material and work with a standard conforming discretization elsewhere. The coupling of different discretization schemes in different subdomains are handled by flexible mortar techniques. A priori error analysis is carried out for the coupled problem, and several numerical examples are presented. Using dual Lagrange multipliers, the Lagrange multipliers can easily be eliminated by local static condensation.

This work is concerned with the design of a $hp$-like discontinuous Galerkin (DG) method for solving the two-dimensional time-domain Maxwell equations on non-conforming locally refined triangular meshes. The proposed DG method allows non-conforming meshes with arbitrary-level hanging nodes. This method combines a centered approximation for the evaluation of fluxes at the interface between neighboring elements of the mesh, with a leap-frog time integration scheme. It is an extension of the DG formulation recently studied in [13]. Several numerical results are presented to illustrate the efficiency and the accuracy of the method, but also to discuss its limitations, through a set of 2D propagation problems in homogeneous and heterogeneous media.

We prove a priori error estimates in a weighted energy norm to the modified method of characteristics (MMOC) for time-dependent convection-diffusion equations with degenerate diffusion. The convergence rates are independent of the lower bound of the diffusion. In other words, these estimates hold uniformly with respect to the degenerate diffusion.

In this paper, we discuss the initial-boundary value problem for the heat transfer equation in lattice-type structures that arises from the aerospace industry and the structural engineering. The main results obtained in this paper are the convergence theorems by using the homogenization method and the multiscale asymptotic method (see Theorems 2.1 and 2.2). Some numerical examples are given for three types of lattice structures. These numerical results suggest that the first-order multiscale method should be a better choice compared with the homogenization method and the second-order multiscale method for solving the heat transfer equations in lattice-type structures.

The impact of parameter dependent boundary conditions on solutions of a class of nonlinear partial differential equations and on optimization problems constrained by such equations is considered. The tools used to gain insights about these issues are the Analysis of Variance (ANOVA) expansion of functions and the related notion of the effective dimension of a function; both concepts are reviewed. The effective dimension is then used to study the accuracy of truncated ANOVA expansions. Then, based on the ANOVA expansions of functionals of the solutions, the effects of different parameter sampling methods on the accuracy of surrogate optimization approaches to constrained optimization problems are considered. Demonstrations are given to show that whenever truncated ANOVA expansions of functionals provide accurate approximations, optimizers found through a simple surrogate optimization strategy are also relatively accurate. Although the results are presented and discussed in the context of surrogate optimization problems, most also apply to other settings such as stochastic ensemble methods and reduced-order modeling for nonlinear partial differential equations.

We analyze analytic approximation formulae for pricing zero-coupon bonds in the case when the short-term interest rate is driven by a one-factor mean-reverting process with a volatility nonlinearly depending on the interest rate itself. We derive the order of accuracy of the analytical approximation due to Choi and Wirjanto. We furthermore give an explicit formula for a higher order approximation and we test both approximations numerically for a class of one-factor interest rate models.

Partial Differential Equations (PDEs) have become an important tool in image processing and analysis. A PDE mode for image zooming is introduced in this paper. This model exploits a higher order nonlinear partial differential equation. The resulted nonlinear equation is solved by an explicit finite difference schemes. Numerical results on real digital images are given to show effectiveness and reliability of the proposed algorithm.

In this paper, a new stable nonconforming mixed finite element scheme is proposed for the stationary conduction-convection problem, in which a new low order Crouzeix-Raviart type nonconforming rectangular element is taken as approximation space for the velocity, the piecewise constant element for the pressure and the bilinear element for the temperature, respectively. The convergence analysis is presented and the optimal error estimates in a broken $H^1$-norm for the velocity, $L^2$-norm for the pressure and $H^1$-seminorm for the temperature are derived.

Image registration has many real life applications. Affine image registration is one of the commonly-used parametric models. Iterative solution methods for the underlying least squares problem suffer from convergence problems whenever good initial guesses are not available. Variational models are non-parametric deformable models that have been proposed based on least squares fitting and regularization. The fast iterative solution methods often require a reliable parametric (affine) method in a pre-registration step. In this paper we first survey and study a class of methods suitable for providing the good initial guesses for the affine model and a diffusion based variational model. It appears that these initialization methods, while useful for many cases, are not always reliable. Then we propose a regularized affine least squares approach that can overcome the convergence problems associated with existing methods. Combined with a cooling idea in a multiresolution setting, it can ensure robustness and selection of the optimal coupling parameter efficiently. Numerical examples are given to demonstrate the effectiveness of our proposed approach.

Adaptive finite element approximation for a constrained optimal control problem is studied. A posteriori error estimators equivalent to the $L^2$ norm of the approximation error are derived both for the state and the control approximation, which are particularly suitable for an adaptive multi-mesh finite element scheme and applications where $L^2$ error is more important. The error estimators are then implemented and tested with promising numerical results.