Exponentially Convergent Parallel Discretization Methods for the First Order Evolution Equations

We propose a new discretization of an initial value problem for differen- tial equations of the first order in a Banach space with a strongly P-positive operator coefficient. Using the strong positiveness, we represent the solution as a Dunford- Cauchy integral along a parabola in the right half of the complex plane, then transform it into real integrals over (−∞,∞), and finally apply an exponentially convergent Sinc quadrature formula to this integral. The integrand values are the solutions of a finite set of elliptic problems with complex coefficients, which are independent and may be solved in parallel.