In this paper, we estimate the error of the linear finite element solutions of the obstacle problem and the unilateral problem with monotone operator. We obtained $O(h)$ error bound for the obstacle problem and $O(h^{3/4})$ error bound for the unilateral problem. And if the solution $u^*$ of the unilateral problem possesses more smoothness, then $O(h)$ error bound can be obtained in the same way as [2].

In this paper, a strategy is suggested for numerical solution of a kind of parabolic partial differential equations with nonlinear boundary conditions and discontinuous coefficients, which arise from practical engineering problems. First, a difference equation at the discontinuous point is established in which both the stability and the truncation error are consistent with the total difference equations. Then, on account of the fact that the coefficient matrix of the difference equations is tridiagonal and nonlinearity appears only in the first and the last equations, two algorithms are suggested: a mixed method combining the modified Gaussian elimination method with the successive recursion method, and a variant of the modified Gaussian elimination method. These algorithms are shown to be effective.

In this paper we prove under certain weak conditions that two classes of implicit difference schemes for the generalized non-linear schrödinger system are convergent and that an iteration method for the corresponding non-linear difference equation is convergent. Therefore, quite a complete theoretical foundation of implicit schemes for the generalized non-linear Schrödinger system is established in this paper.

We have considered the infinite element method for a class of elliptic systems with constant coefficients in [1]. This class can be characterized as: they have the invariance under similarity transformations of independent variables. For example, the Laplace equation and the system of plane elastic equations have this property. We have suggested a technique to solve these problems by applying this property and a self similar discretization, and proved the convergence. Not only the average convergence of the solutions has been discussed, but also term-by-term convergence for the expansions of the solutions. The second convergence manifests the advantage of the infinite element method, that is, the local singularity of the solutions can be calculated with high precision.

In this paper we discuss the estimation for solutions of the ill-posed Cauchy problems of the following differential equation$\frac{du(t)}{dt}=A(t)u(t)+N(t)u(t),\forall t\in (0,1)$, where A(t) is a p. d. o. (pseudo-differential operator(s)) of order 1 or 2, N(t) is a uniformly bounded $H-›H$ linear operator. It is proved that if the symbol of the principal part of A(t) satisfies certain algebraic conditions, two estimates for the solution u(t) hold. One is similar to the estimate for analytic functions in the Three-Circle Theorem of Hadamard. Another is the estimate of the growth rate of ||u(t)|| when $A(1)u(1)\in H$.

The unbounded estimate is one of the troublesome problems in Monte Carlo method. Particularly, in the calculation for the flux at a point, the estimate may approach infinite. In this paper, a collision probability method is proposed in Monte Carlo calculation for the flux at a point, and two kinds of methods with the bounded estimation are presented: the quadratic collision probability method and the importance sampling method. The former method is simple and easy to use, whereas the latter is suitable for calculation of flux at many different points simultaneously. The practical calculation indicates that the variance of the present methods can be reduced by about 50% and the efficiency can be increased by 2 to 4 time in comparison with the existing methods.  

In this paper, a family of 3-dimensional elements different from isoparametric serendipity is developed according to the variational principle and the convergence criteria of the mixed stiffness finite element method. For the new family, which is named mixed stiffness elements, the number of nodes on the quadratic element is not 20 but 14. Theoretical analysis and various computational comparisons have found the mixed stiffness element superior over the isoparametric serendipity element, especially a substantial improvement in computational efficiency can be achieved by replacing the 20 node-isoparametric element with the 14-node mixed stiffness element.