##
**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.

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 |

### Keywords:

contour integral representation; error estimate; parallel computation; convergence; discretization in time; linear parabolic problems; symmetric, positive definite operator; Hilbet space; quadrature; finite element method
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\textit{D. Sheen} et al., Math. Comput. 69, No. 229, 177--195 (2000; Zbl 0936.65109)

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