Simulations

1. Hybridized Discontinuous Galerkin

Please have a look to my blog for more simulations and details

EChapter 1

EChapter 2 

2. Optimized Parallel Computing for 2 Dimensional Heat Equation, 2 Subdomains 

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.

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.

3. Optimized Paralleling Computing for 2 Dimensional Advection-Diffusion Equation, 4 Subdomains

We consider the 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.

4. Optimized Parallel Computing for the Kolmogorov Equation, 2 Subdomains

We let T=2. We decompose the [0,1]x[-1,1] into two subdomains. We discretize the subdomains by a uniform grid. We use dv=dx=dt=0.01. As a consequence, the interface problems features 20,200 unknowns. We choose an overlap of three elements 3dv. We initialize the interface variable with a random value. Finally, we consider the algorithm to have converged when the error drops below 10-6.

We compare the performance of the Classical, 1-sided Optimized Robin and 2-sided Optimized Robin algorithms. We observe the numbers of iterations till convergence of the algorithms in four successive dyadic mesh refinements, 2-jx0.01 with j=0,..,4.

In the overlapping case, both OSWR(p) (1-sided Robin) and OSWR(p,q) (2-sided Robin) algorithms appear to be almost insensitive to the mesh refinement, while the CSWR appears to be very sensitive to it. The two-sided OSWR(p,q) appears globally more robust in terms of iteration counts with respect to the one-sided OSWR(p), whose iteration counts still remain more than reasonable. Both algorithms outperform the CSWR. In the non-overlapping case, a similar pattern is observed for OSWR(p) and OSWR(p,q). Both algorithms appear to be a little sensitive to the size of the interface problem. However, iteration counts are higher than in the overlapping case, but not significantly higher. The OSWR(p,q) is more robust than the OSWR(p), featuring an increase of around 50% in iterations for the most refined case, while the latter experiences a doubling. For both algorithms, however, the iteration counts remain reasonable in all cases. As expected, CSWR does not converge in the absence of overlap.

5. Nonlinear approximation theory for the Boltzmann equation