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domain decomposition method; parareal method; finite element method; heat equation
We present a parallel solution algorithm for the transient heat equation in one and two spatial dimensions. The problem is discretized in space by the lowest-order conforming finite element method. Further, a one-step time integration scheme is used for the numerical solution of the arising system of ordinary differential equations. For the latter, the parareal method decomposing the time interval into subintervals is employed. It leads to parallel solution of smaller time-dependent problems. At each time slice a pseudo-stationary elliptic heat equation is solved by means of a domain decomposition method (DDM). In the $2d$, case we employ a nonoverlapping Schur complement method, while in the $1d$ case an overlapping Schwarz DDM is employed. We document computational efficiency, as well as theoretical convergence rates of FEM semi-discretization schemes on numerical examples.
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