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We study the high-order local discontinuous Galerkin (LDG) method for the$p$-Laplace equation. We reformulate our spatial discretization as anequivalent convex minimization problem and use a preconditioned gradientdescent method as the nonlinear solver. For the first time, a weightedpreconditioner that provides $hk$-independent convergence is applied in the LDGsetting. For polynomial order $k \geqslant 1$, we rigorously establish thesolvability of our scheme and provide a priori error estimates in amesh-dependent energy norm. Our error estimates are under a different andnon-equivalent distance from existing LDG results. For arbitrarily high-orderpolynomials under the assumption that the exact solution has enough regularity,the error estimates demonstrate the potential for high-order accuracy. Ournumerical results exhibit the desired convergence speed facilitated by thepreconditioner, and we observe best convergence rates in gradient variables inalignment with linear LDG, and optimal rates in the primal variable when $1 < p\leqslant 2$.

A high-order local discontinuous Galerkin method for the $p$-Laplace equation