Riemann problem for a geometrical optics system

The paper solves analytically the Riemann problem for a nonstrictly hyperbolic system of conservation laws arising in geometrical optics,in which the flux contains the nonconvex function possessing an infinite number of inflection points.Firstly,the generalized Rankine–Hugoniot relations and entropy condition of delta shock waves and left(right)-contact delta shock waves are proposed and clarified.Secondly,with the help of the convex hull,seven kinds of structures of Riemann solutions are obtained.The solutions fall into three broad categories with a series of geometric structures involving simultaneously contact discontinuities,vacuums and delta shock waves.Finally,numerical experiments confirm the theoretical analysis.     

Regularity for solutions to anisotropic obstacle problems

For Ω a bounded subset of R n,n 2,ψ any function in Ω with values in R∪{±∞}andθ∈W1,(q i)(Ω),let K(q i)ψ,θ(Ω)={v∈W1,(q i)(Ω):vψ,a.e.and v-θ∈W1,(q i)0(Ω}.This paper deals with solutions to K(q i)ψ,θ-obstacle problems for the A-harmonic equation-divA(x,u(x),u(x))=-divf(x)as well as the integral functional I(u;Ω)=Ωf(x,u(x),u(x))dx.Local regularity and local boundedness results are obtained under some coercive and controllable growth conditions on the operator A and some growth conditions on the integrand f.     

自然电位测井的数学模型

自然电位测井是石油开发中一种常用而重要的测井方法。自然电位函数的数学模型可归结为具有间断交界面条件的椭圆型等位面边值问题,在交界面交汇点处,自然电位的跳跃通常不满足相容性条件,此时,这个边值问题不存在分块H1*解,不能使用有限元素法直接求解。本文介绍这一测井方法的数学模型及数值求解方法。     

Spectral analysis for the Gauss problem on continued fractions

We present a new derivation of the formula appearing in Babenko (1978) and Mayer and Roepstorff (1987) that gives the probability distribution of τ⁻ⁿ${\tau }^{-n}$ in terms of the eigenvalues of a symmetric operator. Here τ$\tau$ is the well-known Gauss-map.     

Uniform convergence analysis of hybrid numerical scheme for singularly perturbed problems of mixed type

In this article, we consider a class of singularly perturbed mixed parabolic‐elliptic problems whose solutions possess both boundary and interior layers. To solve these problems, a hybrid numerical scheme is proposed and it is constituted on a special rectangular mesh which consists of a layer resolving piecewise‐uniform Shishkin mesh in the spatial direction and a uniform mesh in the temporal direction. The domain under consideration is partitioned into two subdomains. For the spatial discretization, the proposed scheme is comprised of the classical central difference scheme in the first subdomain and a hybrid finite difference scheme in the second subdomain, whereas the time derivative in the given problem is discretized by the backward‐Euler method. We prove that the method converges uniformly with respect to the perturbation parameter with almost second‐order spatial accuracy in the discrete supremum norm. Numerical results are finally presented to validate the theoretical results.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1931–1960, 2014     

Uniformly convergent C 0‐nonconforming triangular prism element for fourth‐order elliptic singular perturbation problem

In this article, we introduce a C ⁰‐nonconforming triangular prism element for the fourth‐order elliptic singular perturbation problem in three dimensions by using the bubble functions. The element is proved to be convergent in the energy norm uniformly with respect to the perturbation parameter. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1785–1796, 2014