Efficient algorithms for solving the p-Laplacian in polynomial time

The $p$-Laplacian is a nonlinear partial differential equation, parametrized by $p \in [1,\infty]$. We provide new numerical algorithms, based on the barrier method, for solving the $p$-Laplacian numerically in $O(\sqrt{n}\log n)$ Newton iterations for all $p \in [1,\infty]$, where $n$ is the number of grid points. We confirm our estimates with numerical experiments.Comment: 28 pages, 3 figure

An optimal Schwarz preconditioner for a class of parallel adaptive finite elements

A Schwarz-type preconditioner is formulated for a class of parallel adaptive finite elements where the local meshes cover the whole domain. With this preconditioner, the convergence rate of Krylov methods is shown to depend only on the ratio of the second largest and smallest eigenvalues of the preconditioned system. These eigenvalues can be bounded independently of the mesh sizes and the number of subdomains, which proves the proposed preconditioner is optimal. Numerical results are provided to support the theoretical findings