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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 (do not use 65Fxx)
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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