论文标题
具有特征性和指数层的局部不连续的对流扩散问题的局部不连续的Galerkin方法
The local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem with characteristic and exponential layers
论文作者
论文摘要
研究了一个奇特的扰动对流 - 扩散问题，在$ \ mathbb {r}^2 $中提出的对流 - 扩散问题。它的解决方案具有指数和特征边界层。该问题是使用Shishkin网格上的局部不连续的Galerkin（LDG）方法来解决的。在每个变量中，使用张张量的分段多项式，最多$ k> 0 $在每个变量中，LDG解决方案和真实解决方案之间的误差被证明是在单数扰动参数中均匀地收敛的，以$ o（（n^{ - 1}} \ ln n n n n n n n n n n n n n n n n n Nord Normess $ n Nord n osem n Norge n ose n Norge n ose n oserme方向。（这是适用于特征边界层问题的LDG方法证明的第一个均匀收敛结果。）此外，我们证明，这种收敛顺序增加到$ o（（（n^{ - 1} \ ln n n）^{k+1}）当一个人衡量ldg解决方案之间的能量差异和局部元素之间的限制时，统一的超级属性属性意味着我们的LDG方法的最佳$ l^2 $错误估计$（n^{ - 1} \ ln n）^{k+1} $。数值实验显示了我们理论结果的清晰度。
A singularly perturbed convection-diffusion problem,posed on the unit square in $\mathbb{R}^2$, is studied; its solution has both exponential and characteristic boundary layers. The problem is solved numerically using the local discontinuous Galerkin (LDG) method on Shishkin meshes. Using tensor-product piecewise polynomials of degree at most $k>0$ in each variable, the error between the LDG solution and the true solution is proved to converge, uniformly in the singular perturbation parameter, at a rate of $O((N^{-1}\ln N)^{k+1/2})$ in an associated energy norm, where $N$ is the number of mesh intervals in each coordinate direction.(This is the first uniform convergence result proved for the LDG method applied to a problem with characteristic boundary layers.) Furthermore, we prove that this order of convergence increases to $O((N^{-1}\ln N)^{k+1})$ when one measures the energy-norm difference between the LDG solution and a local Gauss-Radau projection of the true solution into the finite element space.This uniform supercloseness property implies an optimal $L^2$ error estimate of order $(N^{-1}\ln N)^{k+1}$ for our LDG method. Numerical experiments show the sharpness of our theoretical results.
