Domain decomposition methods coupled with parareal for the transient heat equation in 1 and 2 spatial dimensions. (English) Zbl 1538.65327
Summary: 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.
MSC:
| 65M55 | Multigrid methods; domain decomposition 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 |
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65F08 | Preconditioners for iterative methods |
| 35K05 | Heat equation |
| 35Q79 | PDEs in connection with classical thermodynamics and heat transfer |
| 65Y05 | Parallel numerical computation |
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