Optimized Parallel Computing for 2 Dimensional Heat Equation, 2 Subdomains

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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].

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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.