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Asymptotics of the Solution to a Stationary Piecewise-Smooth Reaction-Diffusion-Advection Equation

Chinese Annals of Mathematics,Series B 2023年第1期44卷 81-98页

作者： Qian YANG Mingkang NI College of Science University of Shanghai for Science and TechnologyShanghai 200093China School of Mathematical Sciences East China Normal UniversityShanghai 200062China Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice Shanghai 200241China

A singularly perturbed boundary value problem for a piecewise-smooth nonlinear stationary equation of reaction-diffusion-advection type is studied.A new class of problems in the case when the discontinuous curve which separates the domain is monotone with respect to the time variable is *** existence of a smooth solution with an internal layer appearing in the neighborhood of some point on the discontinuous curve is *** efficient algorithm for constructing the point itself and an asymptotic representation of arbitrary-order accuracy to the solution is *** sufficiently small parameter values,the existence theorem is proved by the technique of matching asymptotic *** example is given to show the effectiveness of their method.

关键词： Reaction-Diffusion-Advection equation internal layer Asymptotic method Piecewise-Smooth dynamical system

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A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems

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Communications in Computational Physics 2013年第7期14卷 370-392页

作者： Eric T.Chung Wing Tat Leung Department of Mathematics The Chinese University of Hong KongHong Kong

In this paper,we present an efficient computational methodology for diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion *** is well known that the numerical computation for these problems requires a significant amount of computermemory and ***,the solutions to these problems typically contain a coarse component,which is usually the quantity of interest and can be represented with a small number of degrees of *** are many methods that aim at the computation of the coarse component without resolving the full details of the *** proposed method falls into the framework of interior penalty discontinuous Galerkin method,which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations.A distinctive feature of our method is that the solution space contains two components,namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse *** addition,stability of the method is *** numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems.

关键词： Multiscale problem sub-grid capturing multiscale basis function boundary layer internal layer

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