对流占优的扩散问题的局部间断Galerkin方法

针对具有周期性边界条件对流占优的扩散问题中的二阶导数,引入辅助变量,构造了局部间断 Galerkin(LDG)方法,并给出了方法的稳定性结果和误差估计式.局部间断Galerkin方法是Runge-Kutta 间断 Galerkin 方法的推广,具有高阶精度,能够灵活处理复杂区域,易于处理复杂边界的边值问题,能够有效去除近似解在间断、大梯度处产生的虚假振荡.数值实验说明,当有限元空问取为一次多项式空间时,LDG 方法具有二阶收敛,误差满足理论估计式.该方法可以推广到更高阶的方程,如Korteweg-de Vries方程、重调和方程等.

Local Discontinuous Galerkin Method for Convection-Dominated Diffusion Problems

Local discontinuous Galerkin method(LDGM) for convection-dominated diffusion problems with periodic boundary conditions is constructed.To deal with the derivative of second order,an auxiliary variable is introduced.The stability and error estimations are presented simultaneously. LDGM is an extension of the Runge-Kutta discontinuous Galerkin method with higher accuracy and flexibility,especially for complicated geometries and boundaries.The method enables to effectively remove the spurious oscillations around the discontinuities or strong gradients regions.The numerical experiments show that second order convergence can be obtained when polynomials of degree 1 are chosen and the numerical errors are consistent with the theoretical error estimations.It can be extended to solve partial differential equations with higher order such as KdV equations and biharmonic equations.