Robust Numerical Methods for Nonlocal (and Local) Equations of Porous Medium Type. Part I: Theory

Abstract

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $\partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \;\text{in}\; \mathbb{R}^N\times(0,T),$ where $\mathfrak{L}^{\sigma,\mu}$ is a general symmetric diffusion operator of Lévy type and $\varphi$ is merely continuous and nondecreasing. 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 Lévy operators allows us to give a unified and compact 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 [F. del Teso, J. Endal, and E. R. Jakobsen, C. R. Math. Acad. Sci. Paris, 355 (2017), pp. 1154--1160]. We also present some numerical tests, but extensive testing is deferred to the companion paper [F. del Teso, J. Endal, and E. R. Jakobsen, SIAM J. Numer. Anal., 56 (2018), pp. 3611--3647] along with a more detailed discussion of the numerical methods included in our theory.