We propose a new nonconforming finite element algorithm to approximate the solution to the elliptic problem involving the fractional Laplacian. We first derive an integral representation of the bilinear form corresponding to the variational problem. The numerical approximation of the action of the corresponding stiffness matrix consists of three steps: (i) apply a sinc quadrature scheme to approximate the integral representation by a finite sum where each term involves the solution of an elliptic partial differential equation defined on the entire space, (ii) truncate each elliptic problem to a bounded domain, (iii) use the finite element method for the space approximation on each truncated domain. The consistency error analysis for the three steps is discussed together with the numerical implementation of the entire algorithm. The results of computations are given illustrating the error behavior in terms of the mesh size of the physical domain, the domain truncation parameter and the quadrature spacing parameter.Date: December 20, 2018. 1 arXiv:1707.04290v2 [math.NA] 18 Dec 2018where (·, ·) denotes the inner product on L 2 (R d ) (see, also [3]). We note that for t > 0, (I − t 2 ∆) −1 is a bounded map of L 2 (R d ) into H 2 (R d ) so that the integrand above is well defined for η, θ ∈ L 2 (R d ). In Theorem 4.1, we show that for η ∈ H r (R d ) and θ ∈ H s−r (R d ), the formula (13) holds and the right hand side integral converges absolutely. It follows that the bilinear form a(·, ·) is given by (14) a(η, θ) = c s ∞ 0 t 2−2s ((−∆)(I − t 2 ∆) −1 η, θ) D dt t , for all η, θ ∈ H s (D).

We consider the finite element approximation of fractional powers of regularly accretive operators via the Dunford-Taylor integral approach. We use a sinc quadrature scheme to approximate the Balakrishnan representation of the negative powers of the operator as well as its finite element approximation. We improve the exponentially convergent error estimates from [A.

We study the numerical approximation of a time dependent equation involving fractional powers of an elliptic operator L defined to be the unbounded operator associated with a Hermitian, coercive and bounded sesquilinear form on H 1 0 (Ω). The time dependent solution u(x, t) is represented as a Dunford Taylor integral along a contour in the complex plane.The contour integrals are approximated using sinc quadratures. In the case of homogeneous right-hand-sides and initial value v, the approximation results in a linear combination of functions (zqI − L) −1 v ∈ H 1 0 (Ω) for a finite number of quadrature points zq lying along the contour. In turn, these quantities are approximated using complex valued continuous piecewise linear finite elements.Our main result provides L 2 (Ω) error estimates between the solution u(·, t) and its final approximation. Numerical results illustrating the behavior of the algorithms are provided. arXiv:1607.07832v2 [math.NA]

Abstract. In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e., an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the non-homogeneous forcing problem (with zero initial data) are considered. The solution operator E(t) for the initial value problem can be written as a Dunford-Taylor integral involving the Mittag-Leffler function e α,1 and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here α denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator W (t) and the forcing functionWe develop and analyze semi-discrete methods based on finite element approximation to the underlying (non-fractional) spatial operator in terms of analogous Dunford-Taylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of 1 h. The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying an exponentially convergent sinc quadrature technique to approximate the Dunford-Taylor integral of the discrete operator and is free of any time stepping.To approximate the convolution appearing in the semi-discrete approximation to the non-homogeneous problem, we apply a pseudo midpoint quadrature. This involves the average of W h (s), (the semi-discrete approximation to W (s)) over the quadrature interval. This average can also be written as a Dunford-Taylor integral. We first analyze the error between this quadrature and the semi-discrete approximation. To develop a fully discrete method, we then introduce sinc quadrature approximations to the Dunford-Taylor integrals for computing the averages.