## A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature.(English)Zbl 0936.65109

The discretization in time for linear parabolic problems $u'(t)+ Au(t)= 0,\quad u(0)= u_0,$ with symmetric, positive definite operator $$A$$, defined on a dense subset of a Hilbet space and possessing a compact inverse, has been considered by many authors using different methods. The technique proposed in this paper is based upon the representation of $$u$$ by $u(t)= {1\over 2\pi i} \int_\Gamma e^{-zt} (A- zI)^{-1} u_0dz,\quad t>0,$ with some path $$\Gamma$$ in the right half plane of $$\mathbb{C}$$.
After rewriting the integral as a real integral, the authors apply quadrature rules as e.g. the composed trapezoidal or Simpson rule. For the spatial discretization, a conformal linear finite element method is used. The method is shown to be convergent of order $$O(N^{-2}+h^2)$$ (trapezoidal rule) or $$O(N^{-4}+ h^2)$$ (Simpson rule) for $$t$$ greater than some bound $$\tau$$ and of lower order for $$t\leq\tau$$. Here, $$N$$ denotes the number of time grid points and $$h$$ the spatial discretization parameter. Because of the independence of the spatial finite element problems, the method can be parallelized.
Reviewer: E.Emmrich (Berlin)

### MSC:

 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65J10 Numerical solutions to equations with linear operators 34G10 Linear differential equations in abstract spaces 35K15 Initial value problems for second-order parabolic equations 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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### References:

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