Inexact GMRES: Laplace equation

Submitted for peer review.

Fig. 5: (LaplaceConvergence.pdf)

Convergence of 1st-kind (solid line) and 2nd-kind (dotted line) solvers for the Laplace equation on a sphere, using a GMRES with FMM-accelerated matrix-vecctor products.

Fig. 6: (LaplaceResidualIterations.pdf)

In a test using a sphere discretized with 32,768 triangles, the residual (solid line, left axis) decreases with successive GMRES iterations while the necessary p (open circles, right axis) to achieve convergence drops quickly.

Fig. 7: (LaplaceSpeedupRelaxation.pdf)

Speed-up using a relaxation strategy for three different triangulations of a sphere (N is the number of surface panels), using 1st-kind and 2nd-kind integral formulations. (Multi-threaded evaluator running on 6 CPU cores.)

Fig. 8: (LaplaceRelaxationP.pdf)

Timings for solving a 1st-kind Laplace integral formulation on a sphere discretized with 32,768 panels, using a relaxed GMRES with different initial values of p, compared with a fixed-p solver. The iteration count was capped at 10 for all cases. (Multi-threaded evaluator running on 6 CPU cores.)

Fig. 9: (LaplaceSpeedupTolerance.pdf)

Speed-ups for solving a 1st-kind Laplace integral problem on a sphere discretized with 8,192 panels, as the GMRES solver's tolerance increases; p=10 for all cases. (Multi-threaded evaluator running on 6 CPU cores.)