## Exponentially convergent parallel discretization methods for the first order evolution equations.(English)Zbl 0998.65056

A new discretization of an initial value problem for first order differential equations in a Banach space with a strongly $$P$$-positive operator coefficient is proposed. Using the strong positiveness, the solution is represented as a Dunford-Cauchy integral along a parabola in the right half of the complex plane. Then it is transformed into a real integral. Finally, the authors apply an exponentially convergent Sinc quadrature formula to this integral. The values of the integrand are the solutions of a finite set of elliptic problems with complex coefficients, which are independent and may be solved in parallel.

### MSC:

 65J10 Numerical solutions to equations with linear operators 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 34G10 Linear differential equations in abstract spaces 35K90 Abstract parabolic equations 65Y05 Parallel numerical computation
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