DFR法与DG法解抛物方程和对流扩散方程等价性

Equivalence Between DFR Method and DG Method for Solving Parabolic Equation and Convection-diffusion Equation

研究直接通量重构法(简称DFR)和间断Galerkin法(简称DG)求解抛物方程和对流扩散方程的等价性问题。研究过程分两部分:第一部分对于DFR法和直接间断Galerkin法求解抛物方程的等价性给出两种证明。第一种证明主要用到K点高斯求积具有2K-1阶代数精度。第二种证明主要用到勒让德多项式、拉登多项式和洛巴托多项式的特殊性质。第二部分对于DFR法和局部间断Galerkin法求解对流扩散方程的等价性给出两种证明。主要思想是利用K-1次多项式至多有K-1个不同的零点,从而通过插值法将局部间断Galerkin法所用到的辅助变量直接表达出来。两种方法等价性的证明对于插值法和投影法解偏微分方程的等价性理论做出了进一步完善。

The equivalence between direct flux reconstruction method and discontinuous Galerkin method for solving parabolic equation and convection-diffusion equation is studied.The research process is divided into two parts:the first part gives two proofs for the equivalence of DFR method and direct discontinuous Galerkin method for solving parabolic equations.The first proof mainly uses K-point Gauss quadrature with 2K-1 order algebraic accuracy.The second proof mainly uses the special properties of Legendre polynomials,Radau polynomial and Lobatto polynomial.In the second part,the equivalence of DFR method and local discontinuous Galerkin method in solving convection-diffusion equation is proved.The main idea is that the polynomial of degree K-1 has at most K-1 different zeros,so that the auxiliary variables used in local discontinuous Galerkin method can be directly expressed by interpolation.The proof of equivalence between the two methods improves the equivalence theory of interpolation method and projection method for solving partial differential equations.