Robust numerical methods for local and nonlocal equations of porous medium type. Part I: Theory
Félix del Teso, Jørgen Endal, Espen Robstad Jakobsen · arXiv (Cornell University) · 2018
We develop the theory for monotone schemes of finite difference type for general nonlinear diffusion equations, $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f(x,t) \qquad\text{in}\qquad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}^{\sigma,\mu}$ is a general symmetric diffusion operator of L\'evy type and $\varphi$ is merely continuous and non-decreasing. This theory covers strongly degenerate Stefan problems and the full range of porous medium and fast diffusion equations. Examples of diffusion operators $\mathfrak{L}^{\sigma,\mu}$ are the Laplacian $\Delta$, the fractional Laplacian $-(-\Delta)^{\frac\alpha2}$, $\alpha\in(0,2)$, discrete operators, and combinations of these. The observation that finite difference type operators are nonlocal L\'evy operators, allows us to give a unified, compact, and efficient nonlocal theory for both local and nonlocal, linear and nonlinear diffusion equations. The theory includes well-posedness, stability, equicontinuity, compactness, and convergence of the methods under minimal assumptions including assumptions that lead to very irregular solutions. In other words, the schemes we introduce are robust in the sense that they converge under very unfavorable conditions. Such general results are possible because we are able to work in the largest class of weak solutions where uniqueness is known: distributional solutions in $L^1\cap L^\infty$. As a byproduct of our numerical analysis, we obtain a new and general existence result for the original PDE, a result that was previously announced. We also present some numerical tests, but extensive numerical testing is deferred to a companion paper along with a more detailed discussion of the numerical methods included in our theory.