The approximation of parabolic equations involving fractional powers of elliptic operators

摘要

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 H01(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)−1v∈H01(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 L2(Ω) error estimates between the solution u(⋅,t) and its final approximation. Numerical results illustrating the behavior of the algorithms are provided.