Diffusion Equation With Discontinuous Coefficients Local Discontinuous Galerkin Method

The elliptic and parabolic problems with discontinuous coefficients are often encountered in many physical problems, such as in material sciences and fluid dynamics. It is still a great challenge to solve these problems efficiently and accurately with numerical methods. Because of the discontinuity of the coefficients along the interface, the solution of such a problem has low regularity on the whole physical domain. Due to the low global regularity and irregular geometry of the interface, it seems difficult to achieve high order of accuracy by the classical finite element method.The local discontinuous Galerkin method is an extension of the Runge-Kutta discontinuous Galerkin method. It has many advantages, such as local conservation, easy handling of complicated geometries, allowance of discontinuities, formal high order accuracy, easy realization of parallel and h-p adaptive calculation. These features make it suitable to solve the problems in which the flux is concerned.In this thesis, the local discontinuous Galerkin (LDG) methods are studied for the elliptic and parabolic equations with discontinuous coefficients. The error estimates are given for the numerical solutions and numerical experiments confirm the theoretical results.In chapter 1, we briefly introduce the development of existing numerical methods for the elliptic and parabolic problems with discontinuous coefficients. The history and development of discontinuous Galerkin method are described. The main results about the local discontinuous Galerkin method for interface problems are presented.In chapter 2, a local discontinuous Galerkin method is proposed to solve one dimensional parabolic equations with discontinuous coefficients. The method is proved to be L² stable. When the finite element space uses interpolative polynomials of order k, the convergence rate for the numerical solution of the continuous-time LDG scheme has an order O(h^k+1/2) in L² norm. In the numer- ical experiments, both explicit and implicit time discretization are used to solve the LDG scheme.Chapter 3 discusses the minimal dissipation local discontinuous Galerkin method for the linear elliptic interface problems in two-dimensional convex polygonal domains. The interface may be arbitrary smooth curves. The convergence rates in L²-norm for the numerical solutions of the potential and flux are of orders O(h²|logh|) and O(h|logh|^1/2), respectively. In numerical experiments, the successive substitution iterative methods are used to solve the LDG schemes.In chapter 4, the minimal dissipation local discontinuous Galerkin method is studied to solve the linear parabolic interface problems in two-dimensional convex polygonal domains. The interface may be arbitrary smooth curves. The proposed method is proved to be L² stable and the convergence rate in the energy norm is of order O(h|logh|^1/2). The numerical experiments are given to confirm the theoretical results.Chapter 5 is concerned about the minimal dissipation local discontinuous Galerkin method for the nonlinear parabolic equations with discontinuous coefficients in two-dimensional convex polygonal domains. The interfaces are straight lines. The proposed method is proved to be L² stable and the convergence rate in the energy norm is of order O(h). Numerical experiments show the efficiency and accuracy of methods.