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In this paper, we develop and study algorithms for approximately solving linear algebraic systems: |${{\mathcal{A}}}_h^\alpha u_h = f_h$|⁠ , |$ 0< \alpha <1$|⁠ , for |$u_h, f_h \in V_h$| with |$V_h$| a finite element approximation space. Such problems arise in finite element or finite difference approximations of the problem |$ {{\mathcal{A}}}^\alpha u=f$| with |${{\mathcal{A}}}$|⁠ , for example, coming from a second-order elliptic operator with homogeneous boundary conditions. The algorithms are motivated by the method of Vabishchevich (2015, Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys. , 282 , 289–302) that relates the algebraic problem to a solution of a time-dependent initial value problem on the interval |$[0,1]$|⁠. Here we develop and study two time-stepping schemes based on diagonal Padé approximation to |$(1+x)^{-\alpha }$|⁠. The first one uses geometrically graded meshes in order to compensate for the singular behaviour of the solution for |$t$| close to |$0$|⁠. The second algorithm uses uniform time stepping, but requires smoothness of the data |$f_h$| in discrete norms. For both methods, we estimate the error in terms of the number of time steps, with the regularity of |$f_h$| playing a major role for the second method. Finally, we present numerical experiments for |${{\mathcal{A}}}_h$| coming from the finite element approximations of second-order elliptic boundary value problems in one and two spatial dimensions. [ABSTRACT FROM AUTHOR]

We study the numerical approximation of fractional powers of accretive operators in this article. Namely, if A is the accretive operator associated with a regular sesquilinear form A(.,.) defined on a Hilbert space V contained in L2(Ω), we approximate A-β for β ∊(0, 1). The fractional powers are defined in terms of the so-called Balakrishnan integral formula. Given a finite element approximation space Vh ⊂ V, A-β is approximated by A-β h πh, where Ah is the operator associated with the form A(.,.) restricted to Vh, and πh is the L2(Ω)-projection onto Vh. We first provide error estimates for (A-β - A-β h πh)f in Sobolev norms with index in [0,1] for appropriate f . These results depend on the elliptic regularity properties of variational solutions involving the form A(.,.) and are valid for the case of less than full elliptic regularity. We also construct and analyse an exponentially convergent sinc quadrature approximation to the Balakrishnan integral defining A-β h πhf . Finally, the results of numerical computations illustrating the proposed method are given. [ABSTRACT FROM AUTHOR]