This work formulates and analyzes a new family of discontinuous Galerkin methods for the time-dependent convection-diffusion equation with highly varying diffusion coefficients, that do not require the use of slope limiting techniques. The proposed methods are based on the standard NIPG/SIPG techniques, but use special diffusive numerical fluxes at some important interfaces. The resulting numerical solutions have an $L^2$ error that is significantly smaller than the error obtained with standard discontinuous Galerkin methods. Theoretical convergence results are also obtained.

We use a front tracking algorithm to explicitly construct entropy solutions for the Lighthill-Whitham-Richards traffic flow model with a flow-density relationship that is piecewise quadratic, continuous, non-smooth, and non-concave. The solution is exact if the initial condition is piecewise linear and the boundary conditions are piecewise constant. The algorithm serves as a fast and accurate solution tool for the prediction of spatio-temporal traffic conditions and as a diagnostic tool for testing the performance of numerical schemes. Numerical examples are used to illustrate the effectiveness and efficiency of the proposed method relative to numerical solutions that are obtained using a fifth-order weighted essentially non-oscillatory scheme.

Stochastic action integral and Lagrange formalism of stochastic Hamiltonian systems are written through construing the stochastic Hamiltonian systems as nonconservative systems with white noise as the nonconservative 'force'. Stochastic Hamilton's principle and its discrete version are derived. Based on these, a systematic approach of producing symplectic numerical methods for stochastic Hamiltonian systems, i.e., the stochastic variational integrators are established. Numerical tests show validity of this approach.

The numerical approximation of an elasto-viscoplastic problem is considered in this paper. Fully discrete approximations are obtained by using the finite element method to approximate the spatial variable and the forward Euler scheme to discretize time derivatives. We first recall an a priori estimate result from which the linear convergence of the algorithm is derived under suitable regularity conditions. Then, an a posteriori error analysis is provided. Upper and lower error bounds are obtained.

We study the $n$-simplex nonconforming Crouzeix-Raviart element in approximating the $n$-dimensional second-order elliptic boundary value problems and the associated eigenvalue problems. By using the second Strang Lemma, optimal rate of convergence is established under the discrete energy norm. The error bound is also valid for the eigenfunction approximations. In addition, when eigenfunctions are singular, we prove that the Crouzeix-Raviart element approximates exact eigenvalues from below. Moreover, our numerical experiments demonstrate that the lower bound property is also valid for smooth eigenfunctions, although a theoretical justification is lacking.

Simulations of cardiac defibrillation are associated with considerable numerical challenges. The cell models have traditionally been discretized by first order explicit schemes, which are associated with severe stability issues. The sharp transition layers in the solution call for stable and efficient solvers. We propose a second order accurate numerical method for the Luo-Rudy phase 1 model of electrical activity in a cardiac cell, which provides sequential update of each governing ODE. An a priori estimate for the scheme is given, showing that the bounds of the variables typically observed during electric shocks constitute an invariant region for the system, regardless of the time step chosen. Thus the choice of time step is left as a matter of accuracy. Conclusively, we demonstrate the theoretical result by some numerical examples, illustrating second order convergence for the Luo-Rudy 1 model.

In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thomée (IMA J. Numer. Anal., 2003) to solve the Black-Scholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very efficient for calculating various option prices. Existence and uniqueness properties of the Laplace transformed Black-Scholes equation are analyzed. Also a transparent boundary condition associated with the Laplace transformation method is proposed. Several numerical results for various options under various situations confirm the efficiency, convergence and parallelization property of the proposed scheme.

In this paper, the semi-implicit Euler (SIE) method for the stochastic differential delay equations with Poisson jump and Markov switching (SDDEwPJMSs) is developed. We show that under global Lipschitz assumptions the numerical method is convergent and SDDEwPJMSs is exponentially stable in mean-square if and only if for some sufficiently small step-size $\Delta$ the SIE method is exponentially stable in mean-square. We then replace the global Lipschitz conditions with local Lipschitz conditions and the assumptions that the exact and numerical solution have a bounded $p$th moment for some $p > 2$ and give the convergence result.

An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion two-point boundary value problem with multiple solutions. Its diffusion parameter $\varepsilon^2$ is arbitrarily small, which induces boundary layers. The Schwarz method invokes two boundary-layer subdomains and an interior subdomain, the narrow overlapping regions being of width $O(\varepsilon| \ln \varepsilon|)$. Constructing sub- and super-solutions, we prove existence and investigate the accuracy of discrete solutions in particular subdomains. It is shown that when $\varepsilon \leq CN^{-1}$ and layer-adapted meshes of Bakhvalov and Shishkin types are used, one iteration is sufficient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in $\varepsilon$, where $N$ is the number of mesh intervals in each subdomain. Numerical results are presented to support our theoretical conclusions.

In the present paper, we discuss the superconvergence of the interpolated Galerkin solutions for Hammerstein equations. With the interpolation post-processing for the Galerkin approximation $x_h$, we get a higher order approximation $I_{2h}^{2r-1}x_h$, whose convergence order is the same as that of the iterated Galerkin solution. Such an interpolation post-processing method is much simpler than the iterated method especially for the weak singular kernel case. Some numerical experiments are carried out to demonstrate the effectiveness of the interpolation post-processing method.

A general superconvergence result is established for the stabilized finite element approximations for the stationary Stokes equations. The superconvergence is obtained by applying the $L^2$ projection method for the finite element approximations and/or their close relatives. For the standard Galerkin method, existing results show that superconvergence is possible by projecting directly the finite element approximations onto properly defined finite element spaces associated with a mesh with different scales. But for the stabilized finite element method, the authors had to apply the $L^2$ projection on a trivially modified version of the finite element solution. This papers shows how the modification should be made and why the $L^2$ projection on the modified solution has superconvergence. Although the method is demonstrated for one class of stabilized finite element methods, it can certainly be extended to other type of stabilized schemes without any difficulty. Like other results in the family of $L^2$ projection methods, the superconvergence presented in this paper is based on some regularity assumption for the Stokes problem and is valid for general stabilized finite element method with regular but non-uniform partitions.