The approximation of parabolic equations involving fractional powers of elliptic operators

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 0 1 ( ź ) . 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 ( z q I - L ) - 1 v ź H 0 1 ( ź ) for a finite number of quadrature points z q lying along the contour. In turn, these quantities are approximated using complex valued continuous piecewise linear finite elements.Our main result provides L 2 ( ź ) error estimates between the solution u ( ź , t ) and its final approximation. Numerical results illustrating the behavior of the algorithms are provided.