A Subdivision-based Geometric Multigrid Method

In this paper we introduce a smooth subdi- vision theory-based geometric multigrid method. While theory and efficiency of geometric multigrid methods rely on grid regularity, this requirement is often not directly fulfilled in applications where partial differential equations are defined on complex geometries. Instead of generating multigrid hierarchies with classical linear refinement, we here propose the use of smooth subdivision theory for automatic grid hierarchy regularization within a geometric multigrid solver. This subdivi- sion multigrid method is compared to the classical geometric multigrid method for two benchmark problems. Numerical tests show significant improvement factors for iteration numbers and solve times when comparing subdivision to classical multigrid. A second study fo- cusses on the regularizing effects of surface subdivision refinement, using the Poisson-Nernst-Planck equations. Subdivision multigrid is demonstrated to outperform classical multigrid.

The Effect of Multigrid Parameters in a 3D Heat Diffusion Equation

The aim of this paper is to reduce the necessary CPU time to solve the three-dimensional heat diffusion equation using Dirichlet boundary conditions. The finite difference method (FDM) is used to discretize the differential equations with a second-order accuracy central difference scheme (CDS). The algebraic equations systems are solved using the lexicographical and red-black Gauss-Seidel methods, associated with the geometric multigrid method with a correction scheme (CS) and V-cycle. Comparisons are made between two types of restriction: injection and full weighting. The used prolongation process is the trilinear interpolation. This work is concerned with the study of the influence of the smoothing value (v), number of mesh levels (L) and number of unknowns (N) on the CPU time, as well as the analysis of algorithm complexity.

Analysis of parallel multigrid methods in real-time fluid simulation

The multigrid method has been widely used in computational fluid dynamics (CFD) numerical calculations because of its strong convergence. To achieve real-time simulation of a fluid in computer graphics (CG), the operation efficiency is also a significant factor to consider except for operational accuracy. For this problem, we introduced two multigrid cycling schemes, V-Cycle and full multigrid (FMG). Moreover, we have proposed a simple geometric multigrid method (GMG), and compared with the existing wide application of algebraic multigrid (AMG). All the calculations are the solution of parallel computing of GPU in this paper. The results showed that our approaches have improved the algorithm’s computational speed and convergence time, which prominently enhanced the efficiency of the fluid simulation.