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  3. Analysis of PDEs
  4. Parallel Computing for the 2 D Heat Equati...

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Partial differential Equations, Numerics and Control

Parallel Computing for the 2 D Heat Equation, 2 Subdomains

Minh-Binh Tran

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We consider the heat equation with Dirichlet homogeneous boundary condition on the 2 dimensional domain [-1,1]x [0,1], and the time interval is [0,1].

We discretize the problem with Euler backward scheme in time, finite element in space and use random initial conditions. The mesh size is dx=dy=dt=0.01; the overlapping length is 1 grid points for overlapping algorithms. We plot the errors versus the numbers of iterations of the five Schwarz methods: classical, non-overlapping Robin, overlapping Robin, non-overlapping Ventcell, overlapping Ventcell. We can see that classical < non-overlapping Robin < non-overlapping Ventcell < overlapping Robin < overlapping Ventcell.

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In the above experiment, we plot the errors of the Ventcell overlapping Schwarz methods after 16 iterations for (p,q) in the interval [0,10]x[0,0.5]. The red star in the picture is the theoretical optimized parameter. We choose dx=dy=0.1, and dt=0.01. We can see that the theoretical optimized parameter is very closed to the numerical optimized parameter.

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Eusko Jaurlaritza - Gobierno Vasco ikerbasque - Basque Foundation for Science Bizkaia xede. Bizkaiko Foru Aldundia innobasque - Agencia vasca de la innovación Universidad del País Vasco (UPV/EHU) Ayuntamiento de Bilbao - Bilboko Udala