In mathematical studies of molecular motors, the stochastic motor motion is modeled using the Langevin equation. If we consider an ensemble of motors, the probability density is governed by the corresponding Fokker-Planck equation. Average quantities, such as, average velocity, effective diffusion and randomness parameter, can be calculated from the probability density. The WPE method was previously developed to solve Fokker-Planck equations (H. Wang, C. Peskin and T. Elston, J. Theo. Biol., Vol. 221, 491-511, 2003). The WPE method has the advantage of preserving detailed balance, which ensures that the numerical method still works even when the potential is discontinuous. Unfortunately, the accuracy of the WPE method drops to first order when the potential is discontinuous. Here we propose an improved version of the WPE method. The improved WPE method a) maintains the second order accuracy even when the potential is discontinuous, b) has got rid of a numerical singularity in the WPE method, and c) is as simple and easy to implement as the WPE method. Numerical examples are shown to demonstrate the robust performance of the improved WPE method.

This paper deals with a uniform (in a perturbation parameter) convergent difference scheme for solving a nonlinear singularly perturbed two-point boundary value problem with discontinuous data of a convection-diffusion type. Construction of the difference scheme is based on locally exact schemes or on local Green's functions. Uniform convergence with first order of the proposed difference scheme on arbitrary meshes is proven. A monotone iterative method, which is based on the method of upper and lower solutions, is applied to computing the nonlinear difference scheme. Numerical experiments are presented.

We present a new two-level implicit difference method of $O(k^2 + kh^2 + h^4)$ for approximating the three space dimensional non-linear parabolic differential equation $u_{xx} +u_{yy} +u_{zz} = f(x, y, z, t, u, u_x, u_y, u_z, u_t)$, $0<x, y, z<1$, $t>0$ subject to appropriate initial and Dirichlet boundary conditions, where $h>0$ and $k>0$ are mesh sizes in space and time directions, respectively. In addition, we also propose some new two-level explicit stable methods of $O(kh^2+h^4)$ for the estimates of ($∂u/∂n$). When grid lines are parallel to $x-$, $y-$ and $z-$ coordinate axes, then ($∂u/∂n$) at an internal grid point becomes ($∂u/∂x$), ($∂u/∂y$) and ($∂u/∂z$), respectively. In all cases, we require only 19-spatial grid points and a single computational cell. The proposed methods are directly applicable to singular problems and we do not require any special technique to handle singular problems. We also discuss operator splitting method for solving linear parabolic equation. This method permits multiple use of the one-dimensional tri-diagonal solver. It is shown that the operator splitting method is unconditionally stable. Numerical tests are conducted which demonstrate the accuracy and effectiveness of the methods developed.

The relation between weak and $p$-th mean convergence of numerical methods for integration of some convex, non-smooth and path-dependent functionals of ordinary stochastic differential equations (SDEs) is discussed. In particular, we answer how rates of $p$-th mean convergence carry over to rates of weak convergence for such functionals of SDEs in general. Assertions of this type are important for the choice of approximation schemes for discounted price functionals in dynamic asset pricing as met in mathematical finance and other commonly met functionals such as passage times in engineering.

We clarify the validity of a method that decouples a boundary value problem of biharmonic equation to two Poisson equations on polygonal domains. The method provides a way of computing deflections of simply supported polygonal plates by using Poisson solvers. We show that such decoupling is not valid if the polygonal domain is not convex. It may fail even when the right hand side function is infinitely smooth and supported away from the reentrant corners.

In this paper, an alternative approach for constructing an a posteriori error estimator for non-conforming approximation of scalar elliptic equation is introduced. The approach is based on the usage of post-processing conforming finite element approximation of the non-conforming solution. Then, the compatible a posteriori error estimator is defined by the local norms of difference between the nonconforming approximation and conforming post-processing approximation on the element plus an additional residual term. We prove in general dimension the efficiency and the reliability of these estimators, without Helmholtz decomposition of the error, nor regularity assumption on the solution or the domain, nor saturation assumption. Finally explicit constants are given, which prove that these estimators are robust in suitable norms.

The governing equations of a compositional model for three-phase multicomponent fluid flow in multi-dimensional petroleum reservoirs are cast in terms of a pressure equation and a set of component mass balance equations in this paper. The procedure is based on a pore volume constraint for component partial molar volumes, which is different from earlier techniques utilizing an equation of state for phase fluid volumes or saturations. The present technique simplifies the pressure equation, which is written in terms of various pressures such as phase, weighted fluid, global, and pseudo-global pressures. The different formulations resulting from these pressures are numerically solved; the numerical computations use a scheme based on the mixed finite element method for the pressure equation and the finite volume method for the component mass balance equations. A qualitative analysis of these formulations is also carried out. The analysis yields that the differential system of these formulations is of mixed parabolic-hyperbolic type, typical for fluid flow equations in petroleum reservoirs. Numerical experiments based on the benchmark problem of the third comparative solution project organized by the society of petroleum engineers are presented.

In this paper quintic spline is used for the numerical solutions of the fourth order linear special case boundary value problems. End conditions for the definition of spline are derived, consistent with the fourth order boundary value problem. The algorithm developed approximates the solutions, and their higher order derivatives. It has also been proved that the method is a second order convergent. Numerical illustrations are tabulated to demonstrate the practical usefulness of method.

In this paper we derive asymptotic error expansions for mixed finite element approximations of general second order elliptic problems in three dimensions. And extrapolation method is applied to improve the accuracy of the approximations with the help of the interpolation postprocessing technique. For the cubic domain and uniform partition, with the extrapolation, the accuracy of the mixed finite element approximations can be improved.

We discuss a priori error estimates for a semidiscrete piecewise linear finite volume element (FVE) approximation to a second order wave equation in a two-dimensional convex polygonal domain. Since the domain is convex polygonal, a special attention has been paid to the limited regularity of the exact solution. Optimal error estimates in $L^2$, $H^1$ norms and quasi-optimal estimates in $L^∞$ norm are discussed without quadrature and also with numerical quadrature. Numerical results confirm the theoretical order of convergence.

In this article, we numerically study the regularity loss of the solutions of non-parametric minimal surfaces with non-zero boundary conditions. Parts of the boundaries have non-positive mean curvature. As expected from theoretical results in such geometry, we find that the solutions may or may not satisfy the boundary conditions depending upon the data. Firstly, we validate the numerical study on the astroid and discuss the various kinds of non-regularity characterizations. We provide an algorithm to test the regularity loss using the numerical results. Secondly, we give a numerical estimate of the threshold value of the boundary condition beyond which no regular solution exists. More theoretical results are also given on the approximation by the regularized solution of the non-regularized one. The regularized solution exhibits a boundary layer. Finally, the study is applied to the catenoid for which the exact threshold value is known. The exact value and the computed one are in good agreement.