We study the numerical approximation of a time dependent equation involving fractional powers of an elliptic operator LL defined to be the unbounded operator associated with a Hermitian, coercive and bounded sesquilinear form on H01(Ω). The time dependent solution u(x,t)u(x,t) is represented as a Dunford–Taylor integral along a contour in the complex plane.#R##N##R##N#The contour integrals are approximated using sinc quadratures. In the case of homogeneous right-hand-sides and initial value vv, the approximation results in a linear combination of functions (zqI−L)−1v∈H01(Ω) for a finite number of quadrature points zqzq lying along the contour. In turn, these quantities are approximated using complex valued continuous piecewise linear finite elements.#R##N##R##N#Our main result provides L2(Ω)L2(Ω) error estimates between the solution u(⋅,t)u(⋅,t) and its final approximation. Numerical results illustrating the behavior of the algorithms are provided.
['Andrea Bonito', 'Wenyu Lei', 'Joseph E. Pasciak']
The Approximation of Parabolic Equations Involving Fractional Powers of Elliptic Operators