3 research outputs found
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
∂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
Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory
We develop a unified and easy to use framework to study robust fully discrete
numerical methods for nonlinear degenerate diffusion equations where is a general
symmetric diffusion operator of L\'evy type and 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
are the (fractional) Laplacians and
for , 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.Comment: 34 pages, 3 figures. To appear in SIAM Journal on Numerical Analysi