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dc.contributor.authorDel Teso F.en_US
dc.contributor.authorEndal J.en_US
dc.contributor.authorR. Jakobsen E.en_US
dc.description.abstractAbstract. We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − Lσ,μ[φ(u)] = f(x,t) in RN × (0,T), where Lσ,μ is a general symmetric diffusion operator of L ́evy type and φ is merely continuous and non-decreasing. We then use this theory to prove con- vergence 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 op- σ,μ α are the (fractional) Laplacians ∆ and −(−∆)2 for α ∈ (0,2), erators L discrete operators, and combinations. The observation that monotone finite difference operators are nonlocal L ́evy 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 conver- gence 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 [28]. We also present some numerical tests, but extensive testing is deferred to the companion paper [31] along with a more detailed discussion of the numerical methods included in our theory.en_US
dc.publisherSIAM Journal on Numerical Analysisen_US
dc.subjectNumerical methodsen_US
dc.subjectdistributional solutions,en_US
dc.subjectnonlocal oper- atorsen_US
dc.subjectfractional Laplacian,en_US
dc.subjectStefan problemen_US
dc.subjectfast diffusion equationen_US
dc.subjectporous medium equationen_US
dc.subjectnonlinear degenerate diffusionen_US
dc.subjecta priori estimatesen_US
dc.subjectrobust methodsen_US
dc.subjectmonotone methodsen_US
dc.subjectfinite differencesen_US
dc.titleRobust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theoryen_US

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