Finite Element Approximation for the Fractional Eigenvalue Problem

Abstract

The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order of such convergence. Finally, we perform some numerical experiments and compare our results with previous work by other authors.2010 Mathematics Subject Classification. 46E35, 35P15, 49R05 65N25, 65N30.

Numerical approximations for a fully fractional Allen–Cahn equation

“…Indeed, considering a smooth function d that behaves like δ(x) = dist(x, ∂Ω) near to ∂Ω, all eigenfunctions φ k belong to the space d s C 2s(−ε) (Ω) (the ε is active only if s = 1/2) and φ k d s does not vanish near ∂Ω [20,33]. Moreover, the best Sobolev regularity guaranteed for solutions of (2.8) is [10]).…”

What is the fractional Laplacian? A comparative review with new results

“…\ Ω (28) has been extensively studied in the g ≡ 0 case [69,70,71,72,73], but literature based on the nonzero case is more recent and limited [73,22,74,75]. For the case of zero exterior Dirichlet condition g ≡ 0 in (28), finite element algorithms have been developed in [69], [71], and in particular the adaptive finite element scheme of [72] has been used for the computations of this paper.…”

“…We point out here that theoretical estimates for the convergence rate for finite-volume schemes are not known at the time of this writing. However, we mention that L 2 convergence rates for finite element schemes for such problems have been established in [74,110]. We now study how the value of the horizon parameter affects the behavior of the solution of the nonlocal problem.…”

A Priori Error Estimates for the Optimal Control of the Integral Fractional Laplacian

“…We derive existence and uniqueness results together with first order necessary and sufficient optimality conditions and regularity estimates. For J defined in (1), the fractional optimal control problem reads as follows: Find min J(u, z) subject to the state equation (9) and the control constraints (4). The set of admissible controls is defined by…”

“…Proposition 6 (energy error estimate for graded meshes): Let Ω ⊂ R 2 and s ∈ (1/2, 1). Let u ∈ H s (Ω) be the solution to (9), and let u T ∈ V(T ) be the solution to the discrete problem (28). If T satisfies (32) with µ = 2 and f + z ∈ C 1−s (Ω) then, we have the error estimate…”