Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory

TL;DR

This paper develops a unified framework for analyzing robust numerical methods for nonlinear degenerate diffusion equations, including local and nonlocal cases, with convergence proofs and applications to various equations like porous medium and fast diffusion.

Contribution

It introduces a comprehensive theory for fully discrete numerical schemes for nonlinear degenerate diffusion equations involving general Lévý-type operators, covering both local and nonlocal cases.

Findings

Proved convergence of many numerical schemes for nonlocal and local equations.

Unified theory includes stability, compactness, and minimal assumptions.

Established a new existence result for solutions of degenerate diffusion equations.

Abstract

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}^{\sigma,\mu}$ is a general symmetric diffusion operator of L\'evy type and $\varphi$ is merely continuous and non-decreasing. We then use this theory to prove convergence for many different numerical schemes. In the nonlocal case most of the results are completely new. Our theory covers strongly degenerate Stefan problems, the full range of porous medium equations, and for the first time for nonlocal problems, also fast diffusion equations. Examples of diffusion operators $\mathfrak{L}^{\sigma,\mu}$ are the (fractional) Laplacians $\Delta$ and $-(-\Delta)^{\frac\alpha2}$ for $\alpha\in(0,2)$, discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L\'evy operators, allows us to give a unified and compact {\em nonlocal} theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes stability, compactness, and convergence of the methods under minimal assumptions -- including assumptions that lead to very irregular solutions. As a byproduct, we prove the new and general existence result announced in \cite{DTEnJa17b}. We also present some numerical tests, but extensive testing is deferred to the companion paper \cite{DTEnJa18b} along with a more detailed discussion of the numerical methods included in our theory.