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## Partial differential Equations, Numerics and Control

### Parallel Computing for the 2 D Advection-Diffusion Equation, 4 Subdomains

Minh-Binh Tran

We consider the following advection diffusion equation on the domain [0,1]x[0,1] The initial and boundary data are 0.

The code uses the finite element method to solve the problem and a triangular mesh is used. The solver is GMRES. The discretization steps in space and time are dx=dy=dt=0.01. We look only at the first iteration in time such that T=dt. In our example, there are four subdomains (M=4) and the decomposition in subdomains follows the x - direction. The overlapping length is 2dx. It means that the first subdomain is [0,0.26]x[0,1], the second one is [0.24,0.51]x[0,1], the third one is [0.49,0.76]x[0,1], and the fourth one is [0.74,1]x[0,1].

We consider the performance of the algorithm for several values of Robin parameter p including small and large ones: 1, 2, 10, 20, 55. On the same figure, we also plot the performance of the algorithm with Dirichlet transmission condition. According to this test, the algorithm with Robin transmission conditions reach the errors of 10-6 after at most 9 iterations while the one with Dirichlet transmission conditions needs 15 iterations to reach this error.

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