Félix del Teso

Nicole Cusimano, Félix del Teso, Luca Gerardo-Giorda et al.

In this work, we propose novel discretizations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heat-semigroup formalism. Specifically, we combine suitable quadrature formulas of the integral with a finite element method for the approximation of the solution of the corresponding heat equation. We derive two families of discretizations with order of convergence depending on the regularity of the domain and the function on which the spectral fractional Laplacian is acting. Our method does not require the computation of the eigenpairs of the Laplacian on the considered domain, can be implemented on possibly irregular bounded domains, and can naturally handle different types of boundary constraints. Various numerical simulations are provided to illustrate performance of the proposed method and support our theoretical results.

Félix del Teso

We formulate a numerical method to solve the porous medium type equation with fractional diffusion \[\frac{\partial u}{\partial t}+(-Δ)^{1/2} (u^m)=0.\] The problem is posed in $x\in \mathbb{R}^N$, $m\geq 1$ and with nonnegative initial data. The fractional Laplacian is implemented via the so-called Caffarelli-Silvestre extension. We prove existence and uniqueness of the solution of this method and also the convergence to the theoretical solution of the equation. We run numerical experiments on typical initial data.

Félix del Teso

It is well known from the work of Caffarelli and Silvestre that the fractional Laplacian $(-Δ_x)^{\fracσ{2}}$ for $σ\in (0,2)$ can be obtained as a Dirichlet-to-Neumann map through an extension problem to the upper half space. In this paper we emphasize that the inverse fractional Laplacian $(-Δ_x)^{-\fracσ{2}}$ has a similar property: it can be obtained as a Neumann-to-Dirichlet map via an extension problem to the upper half space. We also show an explicit formula for the solution of the extension problem. Moreover, we deal with powers of a more general class of second order differential operators defined in open subsets of $\mathbb{R}^N$ using the results of Stinga and Torrea. From this characterization we show possible applications among which we mention the numerical analysis of a wide class of nonlinear and nonlocal equations.

Félix del Teso, Jørgen Endal, Espen R. Jakobsen

We develop unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}[φ(u)]=f(x,t) \qquad\text{in}\qquad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}$ is a general symmetric Lévy type diffusion operator. Included are both local and nonlocal problems with e.g. $\mathfrak{L}=Δ$ or $\mathfrak{L}=-(-Δ)^{\frac\alpha2}$, $α\in(0,2)$, and porous medium, fast diffusion, and Stefan type nonlinearities $φ$. By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are $L^p$-stable for $p\in[1,\infty]$, compact, and convergent in $C([0,T];L_{\text{loc}}^p(\mathbb{R}^N))$ for $p\in[1,\infty)$. The first part of this project is given in \cite{DTEnJa18a} and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of \cite{DTEnJa18a} apply and testing the schemes numerically. Our examples include fractional diffusions of different orders, and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems.

Félix del Teso, Jørgen Endal, Espen R. Jakobsen

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^μ[φ(u)]=0$. Here $\mathcal{L}^μ$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function $φ:\mathbb{R} \to \mathbb{R}$ is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain $L^1$-contraction and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations.