Time discretization of parabolic equations by Laplace transformation and quadrature

Abstract: 

We discuss some versions of a numerical method for the discretization in time of an initial value problem for a parabolic equation in a Banach space framework. The method applies a quadrature rule to a contour integral representation of the solution in the complex plane, typically based on Laplace transformation. For each quadrature point an elliptic problem is solved, and this may be done in parallel.

The error bounds obtained may be used in the analysis of fully discrete methods obtained  by application of our time discretization method to a spatially semidiscrete finite element version of an initial-boundary value problem for a parabolic partial differential or a fractional order diffusion equation.