关于二阶导数的不连续Galerkin离散化
On discontinuous Galerkin discretizations of second-order derivatives
AI总结：
本文探讨了局部不连续Galerkin方法在计算二阶导数中的特性，指出其在某些情况下收敛性问题，但时间依赖扩散问题中误差会迅速衰减。
摘要：
本文探讨了局部不连续Galerkin方法在计算二阶导数中的特性，指出其在某些情况下收敛性问题，但时间依赖扩散问题中误差会迅速衰减。
英文摘要：
Some properties of a Local discontinuous Galerkin (LDG) algorithm are demonstrated for the problem of evaluting a second derivative $g = f_{xx}$ for a given $f$. (This is a somewhat unusual problem, but it is useful for understanding the initial transient response of an algorithm for diffusion equations.) LDG uses an auxiliary variable to break this up into two first order equations and then applies techniques by analogy to DG algorithms for advection algorithms. This introduces an asymmetry into the solution that depends on the choice of upwind directions for these two first order equations. When using piecewise linear basis functions, this LDG solution $g_h$ is shown not to converge in an $L_2$ norm because the slopes in each cell diverge. However, when LDG is used in a time-dependent diffusion problem, this error in the second derivative term is transient and rapidly decays away, so that the overall error is bounded. I.e., the LDG approximation $f_h(x,t)$ for a diffusion equation $\partial f / \partial t = f_{xx}$ converges to the proper solution (as has been shown before), even though the initial rate of change $\partial f_h / \partial t$ does not converge. We also show results from the Recovery discontinuous Galerkin (RDG) approach, which gives symmetric solutions that can have higher rates of convergence for a stencil that couples the same number of cells.
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