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Some properties of a Local discontinuous Galerkin (LDG) algorithm aredemonstrated for the problem of evaluting a second derivative $g = f_{xx}$ fora given $f$. (This is a somewhat unusual problem, but it is useful forunderstanding the initial transient response of an algorithm for diffusionequations.) LDG uses an auxiliary variable to break this up into two firstorder equations and then applies techniques by analogy to DG algorithms foradvection algorithms. This introduces an asymmetry into the solution thatdepends on the choice of upwind directions for these two first order equations.When using piecewise linear basis functions, this LDG solution $g_h$ is shownnot 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 inthe second derivative term is transient and rapidly decays away, so that theoverall error is bounded. I.e., the LDG approximation $f_h(x,t)$ for adiffusion equation $\partial f / \partial t = f_{xx}$ converges to the propersolution (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 theRecovery discontinuous Galerkin (RDG) approach, which gives symmetric solutionsthat can have higher rates of convergence for a stencil that couples the samenumber of cells.

On discontinuous Galerkin discretizations of second-order derivatives