In this paper we introduce new characterizations of the spectral fractional Laplacian to incorporate nonhomogeneous Dirichlet and Neumann boundary conditions. The classical cases with homogeneous boundary conditions arise as a special case. We apply our definition to fractional elliptic equations of order s ∈ (0, 1) with nonzero Dirichlet and Neumann boundary condition. Here the domain Ω is assumed to be a bounded, quasi-convex Lipschitz domain. To impose the nonzero boundary conditions, we construct fractional harmonic extensions of the boundary data. It is shown that solving for the fractional harmonic extension is equivalent to solving for the standard harmonic extension in the very-weak form. The latter result is of independent interest as well. The remaining fractional elliptic problem (with homogeneous boundary data) can be realized using the existing techniques. We introduce finite element discretizations and derive discretization error estimates in natural norms, which are confirmed by numerical experiments. We also apply our characterizations to Dirichlet and Neumann boundary optimal control problems with fractional elliptic equation as constraints.For the numerical computation of solutions of (1.5), we rely on well established techniques, see for instance [12,8,7]. It is even possible to apply a standard finite element method especially if the boundary datum g is regular enough. However, the numerical realization of the nonlocal operator (−∆ D,0 ) s in (1.6) is more challenging. Several approaches have been advocated, for instance, computing the eigenvalues and eigenvectors of −∆ D,0 (cf. [39]), Dunford-Taylor integral representation [13], or numerical schemes based on the Caffarelli-Silvestre (or the Stinga-Torrea) extension, just to name a few. In our work, we choose the latter even though the proposed ideas directly apply to other approaches where (−∆ D,0 ) s appears, for instance [13]. Notice that the aforementioned extension of Caffarelli-Silvestre (or the Stinga-Torrea) is only applicable to (−∆ D,0 ) s and not directly to the operator (−∆ D ) s in (1.1).The extension approach was introduced in [17] for R n , see its extensions to bounded domains [19,40]. It states that (−∆ D,0 ) s can be realized as an operator that maps a Dirichlet boundary condition to a Neumann condition via an extension problem on the semi-infinite cylinder C = Ω × (0, ∞), i.e., a Dirichlet-to-Neumann operator. A first finite element method to solve (1.6) based on the extension approach is given in [37]. This was applied to semilinear problems in [4]. In the context of fractional distributed optimal control problems, the extension approach was considered in [3] where related discretization error estimates are established as well.An additional advantage is that our characterization allows for imposing other types of nonhomogeneous boundary conditions such as Neumann boundary conditions (see sections 2.4 and 5) and that it immediately extends to general second order fractional operators (see Section 8).We remark that the diffic...

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The purpose of this work is to introduce and analyze a numerical scheme to efficiently solve boundary value problems involving the spectral fractional Laplacian. The approach is based on a reformulation of the problem posed on a semi-infinite cylinder in one more spatial dimension. After a suitable truncation of this cylinder, the resulting problem is discretized with linear finite elements in the original domain and with hp-finite elements in the extended direction. The proposed approach yields a drastic reduction of the computational complexity in terms of degrees of freedom and even has slightly improved convergence properties compared to a discretization using linear finite elements for both the original domain and the extended direction. The performance of the method is illustrated by numerical experiments.

A linear quadratic Dirichlet control problem posed on a possibly non-convex polygonal domain is analyzed. Detailed regularity results are provided in classical Sobolev (Slobodetskiȋ) spaces. In particular, it is proved that in the presence of control constraints, the optimal control is continuous despite the non-convexity of the domain.

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.