Initially, the solution is smooth and adaptive order enrichment (p-refinement) is performed. In time, however, the solution steepens to a shock, and a combination of mesh refinement and order enrichment (i.e., adaptive hp-refinement) is used.
The results in the table below demonstrate the reductions in execution time achieved using both adaptive hp-refinement and load balancing. The ratio of the average work load per processor to the maximum work load among the processors (the Avg./Max. Work Ratio) indicates how well the computation was load-balanced. By using the unbalanced adaptive hp-refinement method, the execution time was reduced by roughly 50% relative to the fixed-mesh, fixed-order method with comparable accuracy. The adaptive hp-refinement method with load balancing took only 22% as long as a fixed-mesh, fixed-order method with comparable accuracy.
Adaptive hp-refinement Method | Fixed-mesh, Fixed-order Method | ||
No Balancing | With Balancing | No Balancing | |
Global L1-Error | 0.0220026 | 0.0220026 | 0.0218864 |
Avg./Max. Work Ratio | 0.208 | 0.878 | 0.994 |
Total Max. Communication Time | 319.49 secs. | 459.73 secs. | 583.65 secs. |
Total Max. Balancing Time | 0 secs. | 40.99 secs. | 0 secs. |
Total Execution Time | 2909.50 secs. | 1285.78 secs. | 5858.89 secs. |
To solve this problem, we used a discontinuous finite element method and a local load-balancing algorithm on 512 processors of an Intel Paragon. Solutions with comparable accuracy are obtained from a fixed-mesh, fixed-order local finite element method with a 256x128-element mesh and piecewise quadratic polynomials (p = 2) and from an adaptive hp-refinement method with a 32x16-element base mesh and piecewise linear elements (p = 1) initially. Because of the computation efficiency of the adaptive scheme combined with the local load balancing, the adaptive hp-refinement method required only 43% as much time to obtain a comparable solution.