A note on semilinear fractional elliptic equation: analysis and discretization

Abstract

In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order s ∈ (0, 1). We identify minimal conditions on the nonlinear term and the source which leads to existence of weak solutions and uniform L ∞ -bound on the solutions. Next we realize the fractional Laplacian as a Dirichlet-to-Neumann map via the Caffarelli-Silvestre extension. We introduce a firstdegree tensor product finite elements space to approximate the truncated problem. We derive a priori error estimates and conclude with an illustrative numerical example.

Optimal control of fractional semilinear PDEs

“…Proof of Theorem 3.7. The proof of this theorem also follows along the lines of the proof of the corresponding result for the case L = −∆ contained in [8,Theorem 2.9]. It can also be obtained by taking u 2 = 0 and z 2 = 0 in the proof of Proposition 3.10 below and in that case the growth condition (3.4) on f is not needed.…”

“…We further notice that the results of (ii) can be applied to the classical semilinear problems as well. When a ij = δ ij where the latter denotes the Kronecker delta, we developed a complete analysis, including discretization, and error estimates, for (1.1) in [8]. Such an error analysis can be directly applied to (1.1) under the usual assumptions on Ω and the coefficients a ij .…”

“…Such an error analysis can be directly applied to (1.1) under the usual assumptions on Ω and the coefficients a ij . By following the approach of [8] in conjunction with the estimates for the linear problem [2] a similar error analysis can be developed for (1.5).…”

Fractional Operators Applied to Geophysical Electromagnetics

“…In equating (21) with (22) as required by the variational problem statement (19), we see that the coefficient vector ξ T −N − · · · ξ T N + ζ T is common to both the left and right hand sides, and may therefore be divided out, thus leaving a N (L + 2) × N (L + 2) system of linear equations for the unknown coefficients v T −N − · · · v T N + w T which holds for all functions {ξ } , ζ ∈ V h . Upon solution of the linear system, aggregation of the coefficient vectors v according to (12) plus the vector w completes the sum v + w, which we recognize as the discrete, approximate solution to the original differential equation (5). Because the matrix in (21) is complex-valued, large and nonsymmetric, the solution strategy for the linear system equating (21) and (22) must be carefully chosen for scalability and economy of compute resources.…”

Fractional elliptic quasi-variational inequalities: Theory and numerics

“…3) a.e. where T (W ) denotes the solution to(3.2) and U * the solution to the unconstrained problem (3.3).…”