A front-fixing finite element method is developed for the valuation of American put options on zero-coupon bonds under a class of one-factor models of short interest rates. Numerical results are presented to examine our method and to compare it with the usual finite element method. A conjecture concerning the behavior of the early exercise boundary near the option expiration date is proposed according to the numerical results.

In this paper, we consider the discretization of the Stokes problem on domain partitions with non-matching meshes. We propose a hybrid mortar method, which is motivated by a variational characterization of solutions of the corresponding interface problem. The discretization of the subdomain problems is based on standard inf-sup stable finite element pairs and additional unknowns on the interface. These allow to reduce the coupling between subdomains, which comes from the variational incorporation of interface conditions. The discrete inf-sup stability condition is proven under weak assumptions on the interface mesh, and optimal a-priori error estimates are derived with respect to the energy and $L^2$-norm. The theoretical results are illustrated with numerical tests.

In this paper, we discuss a numerical approximation of the first eigenvalue of the $p$-Laplace operator on the sphere $(S^n, g)$ of $\mathbb{R}^{n+1}$.

A singularly perturbed parabolic equation of convection-diffusion type with an interior layer in the initial condition is studied. The solution is decomposed into a discontinuous regular component, a continuous outflow boundary layer component and a discontinuous interior layer component. A priori parameter-explicit bounds are derived on the derivatives of these three components. Based on these bounds, a parameter-uniform Shishkin mesh is constructed for this problem. Numerical analysis is presented for the associated numerical method, which concludes by showing that the numerical method is a parameter-uniform numerical method. Numerical results are presented to illustrate the theoretical bounds on the error established in the paper.

This study presents a new finite element approximation for an optimal control problem ($P$) governed by the heat equation and with end-point state constraints. The state constraint set $S$ is assumed to have an empty interior in the state space. We begin with building a new penalty functional where the penalty parameter is an algebraic combination of the mesh size and the time step. Based on it, we establish a discrete optimal control problem ($P_{h\tau}$) without state constraints. With the help of Pontryagin’s maximum principle and by suitably choosing the above-mentioned combination, we successfully derive error estimate between optimal controls of problems ($P$) and ($P_{h\tau}$), in terms of the mesh size and time step.

In this paper, the penalty method is proposed and discussed for the unsteady conduction-convection problem in two dimensions. In addition, we analyze its time discretization which is based on the backward Euler implicit scheme. Finally, the main results of this paper that optimal error estimates are obtained for the penalty system and the time discretization under reasonable assumptions on the physical data.

In this paper, we present a new approach to obtain the asymptotic expansion and superconvergence for the linear element on Union Jack mesh. First, we construct a generalized interpolation function and its discrete harmonic extension by using the energy embedding method and the method of separation of variables. Then, we present elaborate estimates for the generalized interpolation function and the harmonic extension. Finally, the asymptotic expansion, superconvergence and extrapolation are obtained based on those estimates.

The Rudin, Osher, and Fatemi model [20] (ROF) for image restoration has been extensively studied due to its edge preserving capability, but for images without edges (jumps), the solution to this model has the undesirable staircasing effect. To improve the model, Lysaker, Lundervold and Tai [14] (LLT) proposed a better second-order functional suitable for restoring smooth images but it is difficult to preserve discontinuities for non-smooth images. It turns out that results from convex combinations of ROF model and LLT model can preserve the main advantages of both models (see [16, 9]). In this paper, we first propose an applicable homotopy algorithm based fixed point method for the LLT model. We then propose two new variants of convex combination models. Numerical experiments are shown to demonstrate the advantages of these combination models and the robustness of our homotopy algorithm.

In this article we consider the construction of general isotropic and anisotropic adaptive mesh refinement strategies, as well as $hp$–mesh refinement techniques, for the numerical approximation of the compressible Euler and Navier–Stokes equations. To discretize the latter system of conservation laws, we exploit the (adjoint consistent) symmetric version of the interior penalty discontinuous Galerkin finite element method. The a posteriori error indicators are derived based on employing the dual-weighted-residual approach in order to control the error measured in terms of general target functionals of the solution; these error estimates involve the product of the finite element residuals with local weighting terms involving the solution of a certain adjoint problem that must be numerically approximated. This general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest, as well as providing efficient error estimation. Numerical experiments demonstrating the performance of the proposed adaptive algorithms will be presented.

Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations whose generators coincide with the mass centroids of the respective Voronoi regions. CVTs have become useful tools in many application domains of arts, sciences and engineering. In this work, for the first time the concept of the periodic centroidal Voronoi tessellations (PCVTs) — CVTs that exhibit certain periodicity properties in the Euclidean space — is introduced and given a rigorous treatment. We discuss the basic mathematical structures of the PCVTs and show how they are related to the so-called CVT clustering energy. We demonstrate by means of a concrete example that the clustering energy can lose smoothness at degenerate points which disproves earlier conjectures about the CVT energy being globally $C^2$-smooth. We discuss a number of algorithms for the computation of PCVTs, including modifications of the celebrated Lloyd algorithm and a recently developed algorithm based on the shrinking dimer dynamics for saddle point search. As an application, we present a catalog of numerically computed PCVT patterns for the two dimensional case with a constant density and a square unit cell. Examples are given to demonstrate that our algorithms are capable of effectively probing the energy surface and produce improved patterns that may be used for optimal materials design. The numerical results also illustrate the intrinsic complexity associated with the CVT energy landscape and the rich geometry and symmetry represented by the underlying PCVTs.

In this paper, we consider the problem of computing numerical solutions for stochastic differential equations (SDEs) of Itô form. A fully explicit method, the split-step forward Milstein (SSFM) method, is constructed for solving SDEs. It is proved that the SSFM method is convergent with strong order $\gamma=1$ in the mean-square sense. The analysis of stability shows that the mean-square stability properties of the method proposed in this paper are an improvement on the mean-square stability properties of the Milstein method and three stage Milstein methods.

In this paper, we propose a mixed spectral method for the Navier-Stokes equations in an infinite strip by using generalized Laguerre functions. We establish some results on mixed generalized Laguerre-Legendre approximation, which play important roles in spectral method for fourth order differential equations. A mixed spectral scheme is provided for stream function form of the Navier-Stokes equations. Its stability and convergence are proved. Numeric results demonstrate the efficiency of suggested algorithm.

In this paper, we consider the incomplete interior penalty method for a class of second order monotone nonlinear elliptic problems. Using the theory of monotone operators, we show that the corresponding discrete method has a unique solution. The a priori error estimate in an energy norm is developed under the minimal regularity assumption on the exact solution, i.e., $u \in H^1(\Omega)$. Moreover, we propose a residual-based a posteriori error estimator and derive the computable upper and lower bounds on the error in an energy norm.