论文标题
当地不连续的Galerkin方法的超级熟悉对流扩散问题
Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem
论文作者
论文摘要
在$ \ mathbb {r}^2 $中，在单位正方形上提出的一个奇特的扰动对流扩散问题，其解决方案具有指数边界层，使用本地不连续的Galerkin（ldg）方法来解决，该方法与分段的多项式$ k> 0 $ k> 0 $ k> 0 $ k> 0 $ k> 0 $ k> 0 $ k> 0 $ k> 0 $ k> 0 $ k> 0 $ k> 0 bakhvalov-shishkin-type和bakhvalov-type.on shishkin-type网格网格均不大于$ o（n^{ - （k+1/2）} $准确的能量规范，该能量规范是由弱型表单所引起的能量规范，其中$ n $ n $ n $ mesh Intervals in n. $ n $ MESH INTERCOLALS中使用了$ n $ MESH INTERALES中的方向。 (Note: all bounds in this abstract are uniform in the singular perturbation parameter and neglect logarithmic factors that will appear in our detailed analysis.) A delicate argument is used in this paper to establish $O(N^{-(k+1)})$ energy-norm superconvergence on all three types of mesh for the difference between the LDG solution and a local Gauss-Radau projection of the exact solution into the finite element space.此超级属性属性意味着新的$ n^{ - （k+1）} $绑定了$ l^2 $ bount每种类型网格和问题的精确解决方案之间的$ l^2 $错误；该界限是最佳的（达到对数因素）。数值实验证实了我们的理论结果。
A singularly perturbed convection-diffusion problem posed on the unit square in $\mathbb{R}^2$, whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with piecewise polynomials of degree at most $k>0$ on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type.On Shishkin-type meshes this method is known to be no greater than $O(N^{-(k+1/2)})$ accurate in the energy norm induced by the bilinear form of the weak formulation, where $N$ mesh intervals are used in each coordinate direction. (Note: all bounds in this abstract are uniform in the singular perturbation parameter and neglect logarithmic factors that will appear in our detailed analysis.) A delicate argument is used in this paper to establish $O(N^{-(k+1)})$ energy-norm superconvergence on all three types of mesh for the difference between the LDG solution and a local Gauss-Radau projection of the exact solution into the finite element space. This supercloseness property implies a new $N^{-(k+1)}$ bound for the $L^2$ error between the LDG solution on each type of mesh and the exact solution of the problem; this bound is optimal (up to logarithmic factors). Numerical experiments confirm our theoretical results.
